Definitely not hamsters. I had several growing up and they would learn how to escape their cages and the cats would get them. As an adult, we had ferrets at the same time as cats but we never let the ferrets out when the cats were around. One was always locked up somewhere when the other was out and about. I don't think anything would happen as I don't see cats seeing ferrets as prey but why risk it.
I had ferrets and they played with the younger cats, probably too pesty for older cats though (the ferrets were in a room with a baby gate, so the cats could avoid them if they wanted). I would worry more about the cats than the ferrets, unless the ferret is very old or sick, or the cat is extremely predatory. Ferrets are sharp-toothed little weezels and not afraid of anything.
The rabbits were also in an open-top enclosure and the cats could go in if they wanted, but they generally ignored each other. Regular rabbits are big enough to kick a cat if they have to, but I would worry about dwarf breeds and small bunnies. (Note: ferrets and rabbits definitely don't mix!)
I wouldn't want cats around birds at all. Yes I know the internet is full of cute pictures of cats and birds together, but any kind of fluttering movement can trigger a cat's predatory instincts, and even the smallest cat scratch or bite can kill a bird (from the bacteria in the cat's saliva).
Small rodents are out for the same reason.
So that's about it for cat friends. Dogs are probably the best, although that's dependent on the individual dog of course. Perhaps some kitties are good friends with pigs or horses, but that's hardly practical for most people.
Haha, my neighbor had a farm cat who was good friends with a possum, but, again, not really an option for most people .
Rabbits really depends on yhe personalities of the animals. I've had free range rabbits with cats over the years and some mixes end up with the two species ignoring each other, others with the rabbit dominating the territory and chasing the cat out, and some with the cat trying to kill the rabbit.
If you have a larger rabbit (over 6 pounds, no dwarf or minis) that is raised around cats and cats who are raised knowing rabbits are off limits I wouldn't worry too much. They will stand up for themselves and have plenty of attitude to back it up. There is a risk of the rabbit hurting the cat just as much as the cat hurting the rabbit. If you start with older cats and add a rabbit into the mix it is going to depend more on your cats personality. Over the course of 20+ years, 8 rabbits and 6 cats, my current rabbit is the only one I've worried about. But he was older when we got him and is especially skittish around other animals. My female cat is the only cat out of the current group that I will not leave unsupervised with the rabbit because I am fairly certain she will try to eat him. The boys see him as a friend and aren't interested in a bunny snack. If the rabbit was a little less skittish I might let him give Rocket a good scare so she stops trying to kill him but whatever happened in the first half of his life left scars that cause him to freeze instead of fight.
I'd personally start with the rabbit, or a pair of rabbits since rabbits do best with a buddy, and when they are around 6 months old introduce a kitten to the mix. The kitten will grow up with rabbits who are already old enough and large enough to push back so the social balance will be based on that. But the rabbits will still be young enough to adapt to the cats without fear. Supervise of course and make sure both sides know not to get rough. Remember to spay and neuter rabbits, it not only helps reduce aggression (yes, rabbits can be very aggressive I've had one take a chunk out of another rabbit) but rabbits have very high reproductive cancer rates.
I think the question is more what animal will the cat might get along with. Some cats can't be trusted around any kind of small animal even with supervision. You can have other animals living in the same home, just don't let the cat in the room that has the animal.
I've had cats and rodents at the same time. Oddly, none of the cats cared for the gerbils or mice. I still kept the cats away from the cage / tank when I wasn't around.
Some people have fish and that often doubles as entertainment for the cat. A tank needs a secure lid to prevent the cat from grabbing a snack.
Fish are great! cats love watching fish in the tank, its really enriching for them. just make sure the tank has a lid. With other small animals it is kind of hit or miss. it really depends on both animals demeanor and if they were raised around other animals. I have seen some cats get along really well with birds, rabbits, rats, etc and the total opposite where one harasses or tries to prey on the other. If you want to try something like free roaming a rabbit I would suggest a larger breed to avoid it looking too much like prey and to introduce territory similar to how you would to a new cat coming into the home ( scent soaking, switching rooms, and supervised time together) to get them used to each other.