Kumi Katsuna
Fatal NO FUN! I get real attached to my pokemon so I can only let them go if I reset the game.
There's also Juggernaut. (only one pokemon for battling, no legends, no ubers, 3 HM slaves that cannot be used, catching additional pokemon would be allowed, and healing items, but no extra pokemon for battling but the one.)
There's also Teach, where you have to learn any of the new moves that your pokemon wants to learn.
There's a breed one where you build your team of six from 6 eggs that were traded to you from another of your games, and you don't get to use any other pokemon until you beat the elite four besides those four, which includes no HM slaves unless they were one of the pokemon in the eggs sent over.
And, my personal favorite, "Own2feet" no fly, no bike, only walking, running, and surf. That one is usually combined with one, two, or three of the other runs.