Mammal Orders and Families - how many have you seen?

If you don't mind I'm going to copy the bolding/underlining format you used so bold is wild, underline is captivity and both is both.
Orders:
Order Monotremata (monotremes)
Order Microbiotheria (Monito del Monte)
Order Paucituberculata (shrew-opossums)
Order Didelphimorphia (American opossums)
Order Notoryctemorphia (marsupial moles)
Order Dasyuromorphia (carnivorous marsupials)
Order Peramelemorphia (bandicoots)
Order Diprotodontia (kangaroos, possums, etc)
Order Afrosoricida (tenrecs and golden moles)
Order Macroscelidea (elephant shrews)
Order Tubulidentata (Aardvark)
Order Proboscidea (elephants)
Order Sirenia (sirenians)
Order Hyracoidea (hyraxes)
Order Pilosa (sloths and anteaters)
Order Cingulata (armadillos) - (I have awful luck with these, always a no-show )
Order Dermoptera (colugos)
Order Scandentia (tree shrews)
Order Primates (primates)
Order Rodentia (rodents)
Order Lagomorpha (lagomorphs)
Order Erinaceomorpha (hedgehogs, gymnures)
Order Soricomorpha (shrews etc)
Order Chiroptera (bats)
Order Pholidota (pangolins)
Order Carnivora (carnivores)
Order Perissodactyla (perissodactyls)
[Order Cetartiodactyla (artiodactyls and cetaceans)]
Order Artiodactyla (artiodactyls)
Order Cetacea (whales)

Families:
Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

That was a good thinking exercise, especially with bats and rodents.

 

My list is as follows. The ? symbols indicate that I think I saw a representative of the family, but the memory is fuzzy:


Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)?
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)?
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)?
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)?
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Cetartiodactyla (artiodactyls and cetaceans) - technically this is the order according to current taxonomy but I'm guessing everyone, or almost everyone, will have seen representatives of both artiodactyls and cetaceans so I've split it below into the two former orders.

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)

Family Ziphiidae (Beaked Whales)

 

29, isn't it? Or 28 using Cetartiodactyla?

 

I've seen 24 orders (based on counting Cetartiodactyla as two), including all of those 24 in captivity and 13 of them in the wild as well. Gaps are, predictably for a European, amongst marsupials and colugos. (Captive, Wild, Both).

Order Monotremata (monotremes)
Order Microbiotheria (Monito del Monte)
Order Paucituberculata (shrew-opossums)
Order Didelphimorphia (American opossums)
Order Notoryctemorphia (marsupial moles)
Order Dasyuromorphia (carnivorous marsupials)
Order Peramelemorphia (bandicoots)
Order Diprotodontia (kangaroos, possums, etc)
Order Afrosoricida (tenrecs and golden moles)
Order Macroscelidea (elephant shrews)
Order Tubulidentata (Aardvark)
Order Proboscidea (elephants)
Order Sirenia (sirenians)
Order Hyracoidea (hyraxes)
Order Pilosa (sloths and anteaters)
Order Cingulata (armadillos)
Order Dermoptera (colugos)
Order Scandentia (tree shrews)
Order Primates (primates)
Order Rodentia (rodents)
Order Lagomorpha (lagomorphs)
Order Erinaceomorpha (hedgehogs, gymnures)
Order Soricomorpha (shrews etc)
Order Chiroptera (bats)
Order Pholidota (pangolins)
Order Carnivora (carnivores)
Order Perissodactyla (perissodactyls)
Order Artiodactyla (artiodactyls)
Order Cetacea (whales)

...................................................................................

On that list of families I think I make it 105 total, of which 45 in the wild. Six wild-only, so the next new captive family I see will make the big 1-0-0. (if only I'd managed to see the Ringtail Possum at Hamerton when it was on-show..!). Despite a lot of areas where @Chlidonias had it over on me, Rodentia in particular came to the rescue! His wild figures beat mine all round though!

Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Cetartiodactyla (artiodactyls and cetaceans)

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

 

For me (*=seen in the wild):

Orders (23/29):
Order Monotremata (monotremes)
Order Microbiotheria (Monito del Monte)
Order Paucituberculata (shrew-opossums)
Order Didelphimorphia (American opossums)*
Order Notoryctemorphia (marsupial moles)
Order Dasyuromorphia (carnivorous marsupials)
Order Peramelemorphia (bandicoots)
Order Diprotodontia (kangaroos, possums, etc)
Order Afrosoricida (tenrecs and golden moles)
Order Macroscelidea (elephant shrews)
Order Tubulidentata (Aardvark)
Order Proboscidea (elephants)
Order Sirenia (sirenians)
Order Hyracoidea (hyraxes)
Order Pilosa (sloths and anteaters)
Order Cingulata (armadillos)
Order Dermoptera (colugos)
Order Scandentia (tree shrews)
Order Primates (primates)
Order Rodentia (rodents)*
Order Lagomorpha (lagomorphs)*
Order Erinaceomorpha (hedgehogs, gymnures)
Order Soricomorpha (shrews etc)*
Order Chiroptera (bats)*
Order Pholidota (pangolins)
Order Carnivora (carnivores)*
Order Perissodactyla (perissodactyls)
Order Artiodactyla (artiodactyls)*
Order Cetacea (whales)

Families:
Order Monotremata (monotremes) (1/2)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte) (0/1)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums) (0/1)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums) (1/1)
Family Didelphidae (American opossums)*

Order Notoryctemorphia (marsupial moles) (0/1)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials) (1/2)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots) (0/2)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc) (5/11)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pygmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles) (1/3)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews) (1/1)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark) (1/1)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants) (1/1)
Family Elephantidae (elephants)

Order Sirenia (sirenians) (1/2)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes) (1/1)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters) (2/4)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos) (1/1)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos) (0/1)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews) (0/2)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates) (12/16)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents) (13/33)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)*
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)*
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)*
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)*
Family Muridae (Old World rats and mice)*

Order Lagomorpha (lagomorphs) (1/2)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)*

Order Erinaceomorpha (hedgehogs, gymnures) (1/1)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc) (1/3)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)*
Family Talpidae (moles, desmans)

Order Chiroptera (bats) (3/19)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)*
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins) (1/1)
Family Manidae (pangolins)

Order Carnivora (carnivores) (12/16)
Family Felidae (cats)*
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)*
Family Ursidae (bears)*
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)*
Family Procyonidae (raccoons, ringtails, coatis)*
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)*

Order Perissodactyla (perissodactyls) (3/3)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Artiodactyla (artiodactyls) (9/10)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)*
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales) (2/14)
Family Balaenidae (right whales)
Family Cetotheriidae (Pygmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pygmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

 

23 Orders / 96 Families. Bold the ones I have seen. Most families seen only in captivity, while Phocoenidae, Balaenopteridae, Physeteridae, Talpidae, Rhinolophidae and Vespertilionidae seen only in the Wild.

Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

 

I wonder how many of us have chosen a certain individual of the Family Iniidae.

 

For me:

Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

22/29 on the orders for me. I would love to see more cetaceans and marsupials, but the UK is a hub for nether I guess, so...

75 families, although I didn't put octodont because I wasn't sure I had seen one.

 

I'm on only 22 orders seen in captivity (although as bandicoot were held at London until 1977, I almost certainly saw them as I spent many hours in the Clore Pavilion and Moonlight world in the 70s)
I have only seen 8 in the wild, all in the uk

 

If it is any help, degus are not uncommon in pet shops here.

 

I make it 95 families

 

Ah sorry, forgot that they were octodonts Thanks.
I've seen 76 families then

 

Following TLD’s format for bold and italicised...

Orders: 24/29 seen in captivity, 14/29 seen in the wild, 25/29 seen in total

Monotremata (monotremes)
Microbiotheria (Monito del Monte)
Paucituberculata (shrew-opossums)
Didelphimorphia (American opossums)
Notoryctemorphia (marsupial moles)
Dasyuromorphia (carnivorous marsupials)
Peramelemorphia (bandicoots)
Diprotodontia (kangaroos, possums, etc)
Afrosoricida (tenrecs and golden moles)
Macroscelidea (elephant shrews)
Tubulidentata (Aardvark)
Proboscidea (elephants)
Sirenia (sirenians)
Hyracoidea (hyraxes)
Pilosa (sloths and anteaters)
Cingulata (armadillos)
Dermoptera (colugos)
Scandentia (tree shrews)
Primates (primates)
Rodentia (rodents)
Lagomorpha (lagomorphs)
Erinaceomorpha (hedgehogs, gymnures)
Soricomorpha (shrews etc)
Chiroptera (bats)
Pholidota (pangolins)
Carnivora (carnivores)
Perissodactyla (perissodactyls)
Artiodactyla (artiodactyls)
Cetacea (whales)

---

Families: 94/156 seen in captivity, 40/156 seen in wild, 101 seen in total

Ornithorhynchidae (Platypus)
Tachyglossidae (echidnas)
---
Microbiotheriidae (Monito del Monte)
---
Caenolestidae (shrew-opossums)
---
Didelphidae (American opossums)
---
Notoryctidae (marsupial moles)
---
Myrmecobiidae (Numbat)
Dasyuridae (dasyurids)
---
Thylacomyidae (Bilby)
Peramelidae (bandicoots)
---
Phascolarctidae (Koala)
Vombatidae (wombats)
Phalangeridae (brushtail possums and cuscuses)
Burramyidae (pigmy possums)
Pseudocheiridae (ringtail possums)
Petauridae (striped possums, gliders)
Acrobatidae (feathertail gliders)
Tarsipedidae (Honey Possum)
Hypsiprymnodontidae (Musky Rat Kangaroo)
Potoroidae (rat kangaroos)
Macropodidae (kangaroos, wallabies)
---
Tenrecidae (tenrecs)
Potamogalididae (otter shrews)
Chrysochloridae (golden moles)
---
Macroscelididae (elephant shrews)
---
Orycteropodidae (Aardvark)
---
Elephantidae (elephants)
---
Dugongidae (Dugong)
Trichechidae (manatees)
---
Procaviidae (hyraxes)
---
Bradypodidae (three-toed sloths)
Megalonychidae (two-toed sloths)
Myrmecophagidae (anteaters)
Cyclopedidae (silky anteaters)
---
Dasypodidae (armadillos)
---
Cynocephalidae (colugos)
---
Tupaiidae (tree shrews)
Ptilocercidae (Pen-tailed Tree Shrew)
---
Cheirogaleidae (dwarf lemurs, mouse lemurs)
Daubentoniidae (Aye-aye)
Lemuridae (lemurs)
Lepilemuridae (sportive lemurs)
Indriidae (woolly lemurs, sifakas)
Lorisidae (lorises)
Galagidae (bushbabies)
Tarsiidae (tarsiers)
Callitrichidae (marmosets, tamarins)
Cebidae (capuchins, squirrel monkeys)
Aotidae (douroucoulis)
Pitheciidae (titis, uakaris, sakis)
Atelidae (howlers, spider monkeys, woolly monkeys)
Cercopithecidae (Old World monkeys)
Hylobatidae (gibbons)
Hominidae (apes)
---
Aplodontiidae (Sewellel)
Sciuridae (squirrels)
Gliridae (dormice)
Anomaluridae (anomalures)
Pedetidae (springhaas)
Castoridae (beavers)
Geomyidae (pocket gophers)
Heteromyidae (kangaroo rats, pocket mice)
Diatomyidae / Laonastidae (Laotian Rock Rat)
Ctenodactylidae (gundis)
Bathyergidae (blesmols)
Hystricidae (Old World porcupines)
Petromuridae (Dassie Rat)
Thryonomyidae (cane rats)
Erethizontidae (New World porcupines)
Chinchillidae (chinchillas, viscachas)
Dinomyidae (Pacarana)
Caviidae (cavies)
Dasyproctidae (agoutis)
Cuniculidae (pacas)
Ctenomyidae (tuco-tucos)
Octodontidae (octodonts)
Abrocomidae (chinchilla rats)
Echimyidae (New World spiny rats)
Myocastoridae (Coypu)
Capromyidae (hutias)
Dipodidae (jerboas, jumping mice)
Platacanthomyidae (spiny dormice)
Spalacidae (bamboo rats, mole rats)
Calomyscidae (mouse-like hamsters)
Nesomyidae (Madagascan and African rats and mice)
Cricetidae (voles, hamsters, New World rats and mice)
Muridae (Old World rats and mice)
---
Ochotonidae (pikas)
Leporidae (rabbits)
---
Erinaceidae (hedgehogs, gymnures)
---
Solenodontidae (solenodons)
Soricidae (shrews)
Talpidae (moles, desmans)
---
Pteropodidae (Old World fruit bats)
Megadermatidae (false vampire bats)
Rhinopomatidae (mouse-tailed bats)
Craseonycteridae (Kitti's Hog-nosed Bat)
Rhinolophidae (horseshoe bats)
Hipposideridae (Old World leaf-nosed bats)
Furipteridae (Smokey Bat and Thumbless Bat)
Thyropteridae (disc-winged bats)
Mormoopidae (moustached bats and ghost-faced bats)
Noctilionidae (bulldog bats)
Phyllostomidae (New World leaf-nosed bats)
Mystacinidae (New Zealand Short-tailed Bat)
Myzopodidae (Madagascan sucker-footed bats)
Natalidae (funnel-eared bats)
Molossidae (free-tailed bats)
Emballonuridae (sheath-tailed or sac-winged bats)
Nycteridae (slit-faced bats)
Vespertilionidae (vesper bats)
Miniopteridae (bent-winged or long-fingered bats)
---
Manidae (pangolins)
---
Felidae (cats)
Viverridae (civets, genets)
Prionodontidae (Asian linsangs)
Eupleridae (Madagascan mongooses)
Nandiniidae (African Palm Civet)
Herpestidae (mongooses)
Hyaenidae (hyaenas)
Canidae (dogs)
Ursidae (bears)
Odobenidae (Walrus)
Otariidae (eared seals)
Phocidae (earless seals)
Mustelidae (weasels, badgers, otters)
Procyonidae (raccoons, ringtails, coatis)
Ailuridae (Red Panda)
Mephitidae (skunks, stink badgers)
---
Equidae (horses)
Tapiridae (tapirs)
Rhinocerotidae (rhinoceroses)
---
Suidae (pigs)
Tayassuidae (peccaries)
Hippopotamidae (hippopotamuses)
Camelidae (camelids)
Tragulidae (mouse deer)
Moschidae (musk deer)
Cervidae (deer)
Antilocapridae (Pronghorn)
Giraffidae (Giraffe, Okapi)
Bovidae (bovids)
---
Balaenidae (right whales)
Cetotheriidae (Pigmy Right Whale)
Balaenopteridae (rorquals)
Eschrichtiidae (Gray Whale)
Delphinidae (dolphins)
Monodontidae (Narwhal, Beluga)
Phocoenidae (porpoises)
Kogiidae (pigmy sperm whales)
Physeteridae (Sperm Whale)
Iniidae (Amazon river dolphins)
Pontoporiidae (La Plata Dolphin)
Lipotidae (Baiji)
Platanistidae (Ganges Dolphin)
Ziphiidae (beaked whales)

 

This was a really interesting exercise! As with many others lists, bolded means seen in captivity and underlined means seen in the wild.

Orders: 25/29 seen in captivity, 8/29 seen is the wild, 25/29 in total.

Order Monotremata (Monotremes)
Order Microbiotheria (Monito del Monte)
Order Paucituberculata (shrew-opossums)
Order Didelphimorphia (American opossums)
Order Notoryctemorphia (marsupial moles)
Order Dasyuromorphia (carnivorous marsupials)
Order Peramelemorphia (bandicoots)
Order Diprotodontia (Kangaroos, Possums, etc)
Order Afrosoricida (tenrecs and golden moles)
Order Macroscelidea (elephant shrews)
Order Tubulidentata (Aardvark)
Order Proboscidea (elephants)
Order Sirenia (sirenians)
Order Hyracoidea (hyraxes)
Order Pilosa (sloths and anteaters)
Order Cingulata (armadillos)
Order Dermoptera (colugos)
Order Scandentia (Tree shrews)
Order Primates (Primates)
Order Rodentia (Rodents)
Order Lagomorpha (Lagomorphs)
Order Erinaceomorpha (hedgehogs, gymnures)
Order Soricomorpha (shrews etc)
Order Chiroptera (bats)
Order Pholidota (pangolins)
Order Carnivora (carnivores)
Order Perissodactyla (perissodactyls)
Order Artiodactyla (artiodactyls)
Order Cetacea (whales)

Familes: 97/135 seen in captivity, 17/135 seen in the wild, 99/135 in total.

Order Monotremata (monotremes)
Family Ornithorhynchidae (Platypus)
Family Tachyglossidae (echidnas)

Order Microbiotheria (Monito del Monte)
Family Microbiotheriidae (Monito del Monte)

Order Paucituberculata (shrew-opossums)
Family Caenolestidae (shrew-opossums)

Order Didelphimorphia (American opossums)
Family Didelphidae (American opossums)

Order Notoryctemorphia (marsupial moles)
Family Notoryctidae (marsupial moles)

Order Dasyuromorphia (carnivorous marsupials)
Family Myrmecobiidae (Numbat)
Family Dasyuridae (dasyurids)

Order Peramelemorphia (bandicoots)
Family Thylacomyidae (Bilby)
Family Peramelidae (bandicoots)

Order Diprotodontia (kangaroos, possums, etc)
Family Phascolarctidae (Koala)
Family Vombatidae (wombats)
Family Phalangeridae (brushtail possums and cuscuses)
Family Burramyidae (pigmy possums)
Family Pseudocheiridae (ringtail possums)
Family Petauridae (striped possums, gliders)
Family Acrobatidae (feathertail gliders)
Family Tarsipedidae (Honey Possum)
Family Hypsiprymnodontidae (Musky Rat Kangaroo)
Family Potoroidae (rat kangaroos)
Family Macropodidae (kangaroos, wallabies)

Order Afrosoricida (tenrecs and golden moles)
Family Tenrecidae (tenrecs)
Family Potamogalididae (otter shrews)
Family Chrysochloridae (golden moles)

Order Macroscelidea (elephant shrews)
Family Macroscelididae (elephant shrews)

Order Tubulidentata (Aardvark)
Family Orycteropodidae (Aardvark)

Order Proboscidea (elephants)
Family Elephantidae (elephants)

Order Sirenia (sirenians)
Family Dugongidae (Dugong)
Family Trichechidae (manatees)

Order Hyracoidea (hyraxes)
Family Procaviidae (hyraxes)

Order Pilosa (sloths and anteaters)
Family Bradypodidae (three-toed sloths)
Family Megalonychidae (two-toed sloths)
Family Myrmecophagidae (anteaters)
Family Cyclopedidae (silky anteaters)

Order Cingulata (armadillos)
Family Dasypodidae (armadillos)

Order Dermoptera (colugos)
Family Cynocephalidae (colugos)

Order Scandentia (tree shrews)
Family Tupaiidae (tree shrews)
Family Ptilocercidae (Pen-tailed Tree Shrew)

Order Primates (primates)
Family Cheirogaleidae (dwarf lemurs, mouse lemurs)
Family Daubentoniidae (Aye-aye)
Family Lemuridae (lemurs)
Family Lepilemuridae (sportive lemurs)
Family Indriidae (woolly lemurs, sifakas)
Family Lorisidae (lorises)
Family Galagidae (bushbabies)
Family Tarsiidae (tarsiers)
Family Callitrichidae (marmosets, tamarins)
Family Cebidae (capuchins, squirrel monkeys)
Family Aotidae (douroucoulis)
Family Pitheciidae (titis, uakaris, sakis)
Family Atelidae (howlers, spider monkeys, woolly monkeys)
Family Cercopithecidae (Old World monkeys)
Family Hylobatidae (gibbons)
Family Hominidae (apes)

Order Rodentia (rodents)
Family Aplodontiidae (Sewellel)
Family Sciuridae (squirrels)
Family Gliridae (dormice)
Family Anomaluridae (anomalures)
Family Pedetidae (springhaas)
Family Castoridae (beavers)
Family Geomyidae (pocket gophers)
Family Heteromyidae (kangaroo rats, pocket mice)
Family Diatomyidae / Laonastidae (Laotian Rock Rat)
Family Ctenodactylidae (gundis)
Family Bathyergidae (blesmols)
Family Hystricidae (Old World porcupines)
Family Petromuridae (Dassie Rat)
Family Thryonomyidae (cane rats)
Family Erethizontidae (New World porcupines)
Family Chinchillidae (chinchillas, viscachas)
Family Dinomyidae (Pacarana)
Family Caviidae (cavies)
Family Dasyproctidae (agoutis)
Family Cuniculidae (pacas)
Family Ctenomyidae (tuco-tucos)
Family Octodontidae (octodonts)
Family Abrocomidae (chinchilla rats)
Family Echimyidae (New World spiny rats)
Family Myocastoridae (Coypu)
Family Capromyidae (hutias)
Family Dipodidae (jerboas, jumping mice)
Family Platacanthomyidae (spiny dormice)
Family Spalacidae (bamboo rats, mole rats)
Family Calomyscidae (mouse-like hamsters)
Family Nesomyidae (Madagascan and African rats and mice)
Family Cricetidae (voles, hamsters, New World rats and mice)
Family Muridae (Old World rats and mice)

Order Lagomorpha (lagomorphs)
Family Ochotonidae (pikas)
Family Leporidae (rabbits)

Order Erinaceomorpha (hedgehogs, gymnures)
Family Erinaceidae (hedgehogs, gymnures)

Order Soricomorpha (shrews etc)
Family Solenodontidae (solenodons)
Family Soricidae (shrews)
Family Talpidae (moles, desmans)

Order Chiroptera (bats)
Family Pteropodidae (Old World fruit bats)
Family Megadermatidae (false vampire bats)
Family Rhinopomatidae (mouse-tailed bats)
Family Craseonycteridae (Kitti's Hog-nosed Bat)
Family Rhinolophidae (horseshoe bats)
Family Hipposideridae (Old World leaf-nosed bats)
Family Furipteridae (Smokey Bat and Thumbless Bat)
Family Thyropteridae (disc-winged bats)
Family Mormoopidae (moustached bats and ghost-faced bats)
Family Noctilionidae (bulldog bats)
Family Phyllostomidae (New World leaf-nosed bats)
Family Mystacinidae (New Zealand Short-tailed Bat)
Family Myzopodidae (Madagascan sucker-footed bats)
Family Natalidae (funnel-eared bats)
Family Molossidae (free-tailed bats)
Family Emballonuridae (sheath-tailed or sac-winged bats)
Family Nycteridae (slit-faced bats)
Family Vespertilionidae (vesper bats)
Family Miniopteridae (bent-winged or long-fingered bats)

Order Pholidota (pangolins)
Family Manidae (pangolins)

Order Carnivora (carnivores)
Family Felidae (cats)
Family Viverridae (civets, genets)
Family Prionodontidae (Asian linsangs)
Family Eupleridae (Madagascan mongooses)
Family Nandiniidae (African Palm Civet)
Family Herpestidae (mongooses)
Family Hyaenidae (hyaenas)
Family Canidae (dogs)
Family Ursidae (bears)
Family Odobenidae (Walrus)
Family Otariidae (eared seals)
Family Phocidae (earless seals)
Family Mustelidae (weasels, badgers, otters)
Family Procyonidae (raccoons, ringtails, coatis)
Family Ailuridae (Red Panda)
Family Mephitidae (skunks, stink badgers)

Order Perissodactyla (perissodactyls)
Family Equidae (horses)
Family Tapiridae (tapirs)
Family Rhinocerotidae (rhinoceroses)

Order Artiodactyla (artiodactyls)
Family Suidae (pigs)
Family Tayassuidae (peccaries)
Family Hippopotamidae (hippopotamuses)
Family Camelidae (camelids)
Family Tragulidae (mouse deer)
Family Moschidae (musk deer)
Family Cervidae (deer)
Family Antilocapridae (Pronghorn)
Family Giraffidae (Giraffe, Okapi)
Family Bovidae (bovids)

Order Cetacea (whales)
Family Balaenidae (right whales)
Family Cetotheriidae (Pigmy Right Whale)
Family Balaenopteridae (rorquals)
Family Eschrichtiidae (Gray Whale)
Family Delphinidae (dolphins)
Family Monodontidae (Narwhal, Beluga)
Family Phocoenidae (porpoises)
Family Kogiidae (pigmy sperm whales)
Family Physeteridae (Sperm Whale)
Family Iniidae (Amazon river dolphins)
Family Pontoporiidae (La Plata Dolphin)
Family Lipotidae (Baiji)
Family Platanistidae (Ganges Dolphin)
Family Ziphiidae (beaked whales)

 

My sums are the following:

26 orders (19 in wild, 25 in captivity)
107 families (57 in wild, 95 in captivity)

The toughest wild families I have seen are probably colugos, striped possums/gliders, and galagos. None is particularly challenging, just expensive and time consuming to travel to their ranges from the USA and only out and about at night. I've actually seen galagos in three different countries. Within the USA the most difficult family I've seen without resorting to trapping is moles. They are super common but rarely aboveground. Families that are easy to see for an American that I haven't yet seen in the wild are earless seals, armadillos, and peccaries.