I probably have a greater variety of pets than any member here...
If you're truly looking for something non-traditional, I think a small goat would be a great pet. You'll want to get one shortly after it's born, and you'll want to bottle feed it (vs. waiting until it's weaned). They can make very adorable pets. I know nothing about bathroom training a goat though... we keep ours in stalls in the barn.
A somewhat non-traditional pet would be to have a rabbit. There are a great variety of breeds and sizes available. (dwarfs to Flemish Giants, and possibly a few breeds that are even larger.) They're somewhat common as pets - kept in pens/cages. However, it's uncommon to just have them running around the house. Rabbits are (allegedly) relatively easy to litter train - just like a cat. In a larger pen/cage, they usually make one area a bathroom area.
There are a lot of other animals that you can have as pets... reptiles have no personalities. Fish - again, no personalities. Some birds make decent pets and actually do have a bit of a personality that you can enjoy. Research the various types of birds first though. Some birds that are common pets (cockatiel) are really cool, easy to breed, but make a huge freakin mess with seeds and stuff - you can't keep their mess contained to their birdcage. You'll be constantly cleaning up the area around the birdcage.