Is it possible to somehow "copy" an NES game that you've compiled onto a real NES cartridge so that it can be played on a real NES? That could make this an incredibly fun hobby. Especially as a parent of kids who actually enjoy NES and SNES games.
The magical thing to google if you want to do your own research is "NES flash cart".
The "EverDrive" that aji suggests is, indeed, an example of a flash cart. It may or may not be the best one for the NES - I honestly don't know, all of my experience is with the GameBoy.
dpflan's approach of erasing and reprogramming a REAL NES cart is also possible, if you enjoy pain and suffering in the name of being lavishly historically accurate. (I know I do sometimes.)
[1] https://en.wikipedia.org/wiki/Flash_cartridge
[2] https://en.wikipedia.org/wiki/Memory_management_controller