Read this: Pet a Tiger Cub?
Do not want to own an ocelot, fox or owl.
Want to adopt a shelter cat or dog, and train it.
I mean... I get so envious watching these people petting tigers and lions, but I'd rather have plenty of tigers and lions living a normal big cat life in the wilderness than knowing that people wanting "unusual pets" causes thousands of animals being abused, by breeders who just want money, by people catching wild animals just to get something valuable to sell, by owners who don't know enough about the animal and its lifestyle to give it proper and adequate care, by vets who do horrible things because they get paid...
You can, of course, get animal tender / zoo keeper / veterinarian education, and get a work at a zoo or a wildlife sanctuary, and that way be able to say you have petted a wild animal.