I’ve played just about every wild character concept conceivable in Diablo II, including the singing barb (only attack being yelling at people; you basically end up being limited by how fast you can digest mana potions, since you’re continually drinking them), melee sorc (do it right, with really good gear, and you can do an average of over 30k damage per hit, at five hits per second) and John Henry (paladin who never uses any attack skill except Sacrifice, and uses the Steeldriver as his weapon). I must admit, though, that I’ve never yet (and probably never will) run Irene the Infirm.
I’ve also done a variety of self-imposed restrictions in the original Final Fantasy, including but not limited to never grinding, and playing with a party of four White Mages. I also always refuse to let my party members’ XP get out of synch, which means that if anyone dies, I either run from everything until I have a chance to res them, or I reload.
In Civilization, one popular limitation is the One City Challenge: You’re only allowed to ever build or otherwise control one city, for the entire game. It’s actually easier than you’d think, provided you choose the right victory condition, and it goes a lot quicker than a regular game. Another restriction is Always War, where you have to immediately declare war with everyone you meet, and never make peace, though I’ve never done that one.
In good old Minesweeper, I usually either don’t mark any mines at all, or mark all mines and clear spaces only by using the left-right-click “clear all around this space” function.
Finally, there’s a classic DOS-era game called Sherlock that I play a lot. It’s basically those logic puzzles with clues like “The man who owns the dog lives in the house next to the violin player”, but in a standardized form and with randomly-generated puzzles. After you’re done with a clue, you can right-click it to make that clue disappear, so as to reduce clutter. Well, I restrict myself to never actually using information from a clue until after I’ve disappeared it, to force myself to remember them.