Everyone likes a good challenge, but some games push the boundaries of sanity. What are some titles you love in spite of (or because of) their extreme difficulty? These need not be games seen as universally impossible, just to you. Hell, if you just can't get your thumbs around a hadouken, vent away!
The first that springs to mind for me is..
Spider-Man/X-Men: Arcade's Revenge, SNES
Good god, this game has bested me for years. I would keep coming back to Arcade's Revenge throughout my childhood in hopes of beating it, gradually working out strategies for each character's stages, but doing the whole thing start-to-finish on a console stands as a feat practically unimaginable.