The answer is found in the relevant state law: Chapter 959: OFFENSES RELATING TO DOMESTIC ANIMALS. I have read the page and there are no restrictions on the number of cats a person can own in Ohio. So there is no limit. However, there are many clauses concerning cruelty to animals and prohibitions concerning companion animals.
If an Ohioan is able to provide for a large number of cats to a good standard and if that person does not cause a nuisance to others then theoretically any number of cats can be owned by a citizen of Ohio provided there are no related laws which could be breached under these circumstances.
For instance there may be laws governing cat breeders and a cat breeder may have lots of cats. That person would have to make sure that they are complying with laws other than Chapter 959.
Chapter 959 refers to both ‘domestic animals’ and ‘companion animals’. I believe these are interchangeable. It includes pets in a pet store. It does not include wild animals and I don’t believe that this law covers tame wild animals considered to be pets in people’s homes such as servals and cougars.
Click this link for Chapter 959.