Which cats? All 36 species!!! Of course impossible, but the more the better as far as I'm concerned. All of the BIG cats are common in U.S. (even pumas, which are almost all rescued from orphaned cubs and are not being bred by AZA facilities). But the smaller cats, which is the majority of cat species, are under-represented. AZA has targeted 8 small cats they want to maintain, which is good and bad. Bad, because it's only 8 and the others are being phased out, but Good because more zoos are being asked to house some of these in order to maintain a self-supporting population.
Here are best U.S. zoos for cats (in terms of many species, at least 8 or more): CINCINNATI ZOO, EXOTIC FELINE BREEDING COMPOUND, THE LIVING DESERT, MEMPHIS ZOO, OKLAHOMA CITY ZOO (although I haven't been), SAN DIEGO ZOO (which has the most of everything, not just cats), PROJECT SURVIVAL CAT HAVEN, BIG CAT RESCUE.
Best cat collection in the world (I believe) is LE PARC DES FELINS in Nesle, France. I will be going there this October (as well as Port Lympne, England) on my first trip outside of the U.S.