In order to keep exotic animals as pets, some states require a permit. Each state may have varying sets of conditions that need to be met to earn said permit. Not to mention that there are different types of permits, such as a Wildlife Hobby Permit, Annual Animal Hobby Permit or Class II Annual Permit.

Level 1: Slightly quirky
In the states of Missouri and Michigan, the following native animals are able to keep as pets so long as a Wildlife Hobby Permit is possessed:
Badgers
Coyotes
Gray and Red Foxes
River Otters
Raccoons

Level 2: Hm. That is…interesting
In the state of Montana, the following animals do not require a permit to keep:
Alpacas
Ostriches
Tammar Wallabies
Toucans
Pythons
Iguanas

Also, the state of Arizona allows capybaras to be kept as pets.

Level 3: That is wild
In the state of Texas, with a Certificate of Registration to own “dangerous wild animals,” these animals can be kept as pets:
Bears
Lions
Gorillas
Cheetahs
Tigers
Orangutans

In Arizona, even zebras and elephants can be kept as pets without a permit of any kind, all according to findlaw.com.