Playing to lose? Absolutely! Some of the best times I've had in a game were when I played to lose!
-Playing Halo 3 splitscreen with a buddy and going online. In our heads, we were always on our own 2-man team, even when the game said we were enemies.
-Dark Souls, currently doing a playthrough where I'm not allowed to block ever. The parrying dagger and bandit knife are my babies.
-Far Cry 2, for a while I was only allowed to use a machete. That led to some of my favorite moments in that game, and there's nothing quite like clearing out an entire encampment and having them shit themselves in fear because of some looney hiding in the foliage with a machete.
-Playing through Cave Story in a minimalist run, meaning I only get my starting 3 HP and the weapons that I absolutely HAVE to collect to proceed with the game.
-Dungeons of Dredmor, rolled a character that was only allowed to carry shields in either hand. Ironically he turned out to be one of my best characters.
-Super Smash Bros, any iteration, at least one person absolutely HAD to use Jigglypuff and everyone has one life. Whoever loses to Jigglypuff has to be Jigglypuff until they beat someone else.
-Playing Kirby's Adventure, picking a power, and only being able to use that power for the whole game.
-Playing D&D and rolling a wizard with high STR, CON, and DEX and virtually nothing in WIS, INT, or CHA. Also in D&D, rolling a kobold barbarian (who turned out to be surprisingly effective).
-Crackdown 2 multiplayer. We were only allowed to use throwable objects and melee combat to kill anything. That was pretty special.
-Trying to do a Batman: Arkham Asylum playthrough on Hard difficulty without upgrading anything was fun as all hell, I'll probably get around to doing that with Arkham City as well.