Hmm... I believe that, if you don't include games I've only rented, there are only two games I've tried to beat and then didn't. They are:
-I wanna be the guy: I'm at the very final battle for this one. Thing is, I got distracted by new games I got (don't remember which) and kind of let it aside. Now it's been over a year and I don't think I have much chance of beating it without a LOT of extra practice. I'd start over, but as much as I loved the game, I don't feel like the maddening frustration will be as interesting the second time around, especially since I won't remember most of the traps.
-Harvest Moon: A Wonderful Life: The biggest disappointment out of any game I've ever had. I've been a fan of the series since I was 9 (I can't help it. I just love it and I'm not too sure why) and I was quite excited about all of the new ideas that were being thrown into the game. I even preordered it to make sure I would get it as soon as it came out. It turned out to be the blandest game in the series, with very little to do past the third year. And they want you to play for 10? (I'm aware that later releases of the game shorten it, but even then...)
I'm also on a temporary hiatus of Professor Layton and the Curious Village. I've beaten it, but the two last puzzles (134 and 135) I still haven't solved. I got frustrated, so I took a break and will get back to them later, as that usually helps. I will not let that game beat me.
You could also count Majora's Mask in a different sense; I've beaten it once, and though I love it and have attempted at replaying it many times, I've never re-beaten it.