The gay men's health center in my city of residence offers the OraQuick rapid test with results in 20 minutes. They also collect blood samples for the longer test for which it takes 2 weeks to get results.
I went to get such a test taken 2 weeks after possible exposure to HIV, and the rapid test did not detect any antibodies. The health center will give you the rapid test so long as it's at least 2 weeks, so I went on the 2 week mark.