Deadly Premonition, multiple times. :P
As for...broken, broken, to the point of game-breaking and still pushing me through, perhaps Fallout:New Vegas. I never found anything game-breaking, as in a mission not being finished or so, but more often than not I would glitch through walls, important NPCs would vanish for weeks, dialogues would repeat themselves for no apparent reason, I would fall into oblivion which caused frustrating deaths, I'd open a door only to be stuck forever in the loading screen, seeing enemies with a flying sniper rifle in front of them using the punch animation to shoot at me...errr...I don't know what else, but there are tons of other things.
I did it because I found the game extremely enjoyable. Heck, it is one of my favourite games ever made.
