I already mentioned this in the introduction thread, but...
Contra - I used to be good enough at Contra that I never needed the Konami code. My greatest gaming accomplishment is beating it twice in a row, including the harder second time through, without dying. I haven't tried it in years, though, so I'd bet that game would destroy me if I tried now
TMNT (arcade) - I would play this game incessantly at the bowling alley where my parents had their league night, and I had every inch of it memorized. To this day I can usually beat it on one credit, but I still have not accomplished my ultimate goal, which is to beat it without dying. When you're playing 1-player, the longer you go without dying, the game throws more enemies at you and the bosses take more hits, until you finally die and the enemy count is reduced back to normal. Around the time you get to the middle of stage 3, the number of enemies becomes almost overwhelming if you haven't died, making it extremely difficult for me to get through it without losing one life. One time I got to Krang on one life, but he was taking an INSANE amount of hits to beat and I got careless and he killed me.
I'm pretty good at games similar to TMNT, like Simpsons and X-men, but those games have some cheap bosses that make it very difficult to beat them without multiple deaths (for example, Blob at the end of X-Men stage 2, who sometimes does an unavoidable throw that takes off a chunk of life).
Q*bert - I'm good enough at Q*Bert (my #1 favorite game ever) that I can be out of practice and still do well. Last October I went to a retro arcade in Austin, Texas during a business trip and proceeded to spend the majority of my time on one game of Q*Bert. I think I got to the point where level 9 was repeating for the 5th or 6th time when I just wanted to play something else, as my time and money were limited. If I had the time, money, and patience to practice for multiple hours a day, I honestly feel like I could possibly challenge for some kind of record.