Depends what you mean by limit but I'm choosing to interpret it as "exert undue influence over the choice of the parent". The UK's restrictions on names containing obscenities, numerals, misleading titles, or are impossible to pronounce seems reasonable although you could make arguments against all of those e.g. some words or names in other languages are obscenities in English and vice versa, numerals are traditionally used as a suffix to distinguish generations with the same name so why not be able to be Nathan3421, misleading titles is a bit stuffy and incongruous with existent names like Major, Judge, Prince, etc., impossible to pronounce already describes many names if you are not familiar with the native language. After all of that, I honestly don't think there is a good reason to restrict a name. Maybe the only limit I could think of would be to prove that it is given in good faith and not for some malevolent purpose.

I think most US states only require the name to fit in their database. They may also require Latin based alphabets or be even just English alphabet with no extra marks and may ban some punctuation or numerical characters. But probably no other restrictions

You probably can't pick obscenity in the US but otherwise anything goes. (Semantically/phonetically at least - orthography is a bit more limited.)

That most European countries have a whitelist of names was one of the more shocking parts of moving here.