Thought I'd try this again, five years later...
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)
Should be 23 orders and 90 odd families. Fingers crossed will be able to tick another order off this summer

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