My tooth fell out and now I'm going to get a flipper. Am I allowed to get braces on top of a flipper?

You likely won't be able to wear your flipper once you have braces since the flipper attaches securely to your teeth and prevents them from moving.  Along the same lines, once your teeth start to move, the flipper won't fit correctly, so your pre-braces flipper won't be able to be worn after braces.  But don't worry, most orthodontists are happy to place a "fake" tooth on the wire so you don't have to go without a tooth for too long.  If Invisalign is an option for you (you can discuss this with your doctor as not all patients are good candidates for aligners) they can create a tooth in the aligners so you'll only be "toothless" when the aligners are taken out.  Once your orthodontics is complete you can talk with your general dentist about the options for a new removable flipper or a fixed (glued in) tooth replacement so you can always feel confident showing off your new smile!