In Australia, you can't get more exotic than a rat...
but, uh...do you want a mammal- becaus eflemish Ginat Rabbits are rather rare and HUGE.
Hedgehog, a skunk? Squirrel?
Depends where you live. SOme places allow foxes.
or, reptilian- like many lizards.
How about a Mexican Walking Fish (axilotal, a larval salamander)
I know a pet-shop that sells stingrays, sand sharks and small ribbon eels. Very cute, I totally want a stingray, they are kinda sweet in their own "flappy floaty flappa, glide" kinda way.