Assuming that the American passport will be the same as those with a UK passport, it can only be renewed once then you must do a 'visa run' (as already mentioned, Bahrain is a good choice - Bahrain Air is inexpensive). Upon arrival back in Qatar you can buy another visit visa for a month which is renewable once only again, and so the cycle restarts. Be careful not to allow the visa to expire before you fly out and back, I believe the fine is 500Qr per day.