Good question. I betook myself to the American Kennel Club's website to find an exact number.
I didn't find that, but I did find a long and expanding list of Recognized Breeds, Miscellaneous Class Breeds, and Foundation Stock Service Breeds. Those last two are, respectively, "breeds in waiting" for AKC official breed status, and an official recordkeeping arrangement for rare breeds not at present formally recognized.
For example, as of today the Redbone Coonhound is a Miscellaneous Class Breed. I'm surprised at that: one of the best dogs my family ever had was a Redbone Coonhound. Note that its temperament is "a good family dog" and "likes to please". Absolutely. (RIP, Moon.)
As for a Foundation Stock Service Breed, hold on to your hat: did you know the Russell Terrier is in this list? Yes, Eddie the dog on "Frazier".
Personally I'm a big fan of mutts (remember to spay and neuter, everybody). But the dog that kicks around the Pet Museum office is a Brittany.