I'm a falconer so I've let many raptors into my home and vehicle. Birds have very sensitive lungs so I do give a lot of thought to air quality in those spaces though. No idling cars or near them for a length of time, no candles, no teflon, that kind of stuff.
Had a neighbor's mini donkey escape last year and found it grazing outside our fence line. She lived in our front yard for several days while we searched for her owner. She was a Super chill animal who liked to stand inside a dog kennel we had setup.
Growing up, some friends had a house kangaroo. It was afraid of everyone but them. Sometimes when visiting I'd find it on the couch and it'd sorta have a low key "oh fuck people" reaction, get up and hop off.
Edit: There's a nice story of one night the old bloke saw the roo grab and drag the labrador out of the warm spot in front of the fire only to steal the spot for itself.
These people also had a donkey that was welcome in the house.
So I'm scared of spiders, but if they're chill I'm chill too. So I just let them be and they usually leave me alone. And I hate it when somebody kills a spider, just want somebody else to take the big ones outside for me :-)
I have a rule in my house. Do not kill spiders. Reason: If there is a spider, there is something I don't want in my house, it is its job to get rid of it. When the food is gone, so is the spider.
Probably a capybara, or a cheetah. Both are rather chill animals, and cheetahs are fairly able to be domesticated (and have been multiple times throughout history) at least compared to other big cats.
Both are still a really bad idea, and the latter will probably result in injury or death. After having interacted with two Cheetahs through a rehab centre, I'd be open to try.
To be fair, "domesticated" cats are as well and no doubt to the same degree. It's just that due to their size they're not in a position to do much to you.
I certainly get randomly attacked by my cats whenever they get a bee in their bonnet, or want something, or are bored, or because it's Tuesday, etc. The rest of the time they're chill.
This makes the scenario seem like the exception rather than the rule, and I find this weird and disturbing.
The fact that most responses go "this one animal" as an answer instead of starting from the genus down makes me think I've either misunderstood something or we're really not on the same page here.
All the cats around us are mostly feral and won't even come close. Letting them in would result in piss, shit, and marking everywhere. I happily try to feed them some treats here and there, but no way they're coming inside the house.