For mammals:
- many primates: Common Marmosets, capuchins, guenons, baboons, macaques
- mustelids and viverrids
- hyenas
For herps:
- Green Iguana
- Yellow Anaconda (AZA is predominantly Green)
- Nile Crocodile
- Common pet store herps (bearded dragons, leopard geckos, ball pythons, etc)
- some monitor lizards (mostly those common in private trade)
Hard to say with birds; it seems like many non-AZA facilities are often lacking in birds generally, with a few exceptions (ratites, peafowl, guineafowl, parrots).