Depends on how interesting the games are. I've played about 6 before.
What is the MAXIMUM amount of rematches you'd give on a losing streak?

Some people just want to increase their win streaks or stats regardless of how interesting the games are. So if they're crushing an opponent, they'll keep offering rematches because they know they'll win again.

If I lose to the same opponent 3 times, I'm done...
Random seeks? Two games and I'm moving on. With a friend I would go longer, but just depends on why I'm losing. If I'm in bad form I may decide to try again another time after a few games.

With random seeks I may play a bit more if they go with different openings. It's very subjective for me, though. If I found a loss legitimately interesting, I'll play again. I recognize that ratings go up and down, but finding that game that really helps you learn is rare. So replaying is sometimes worth a shot.
Beating my opponent playing the same opening over&over, or getting beat by the same opening over&over, isn't good for learning. But if someone is willing to start e4/d4/c4/Nf3 in successive games, I will, too.
If I lose to the same opponent 3 times, I'm done...