What are pets that aren't actually "pets"?

My apartment has a one pet rule, and that is filled by my cat.

What are some creatures that wouldn't technically count as "pets"? Ones like a moss ball, aquarium plants/coral, or perhaps a tarantula/bug?

Note: I do have enough space to make an encloser large enough to make them live happily if that's a concern. I also have done extensive research on the ones I listed and this is not an impulse pet buy