Man, this is somewhat hard as I have so many.
First off, managing to beat the first Teenage Mutant Ninja Turtles game for the NES was no small feat. I think it was more of luck than skill as I attempted to do it again only to get creamed by Shredder. Beating Battletoads was another one, though the most trouble I had with this game was that snake pit level but I persevered. By the way, both of their endings sucked. With Battletoads, I was hoping to be able to play as Pimple, the third Battletoad, after beating the game but I was only disappointed.
Mega Man 4 will always have a place in my heart as defeating Dr. Wily and listening to the epilogue song overwhelmed me with a sense of accomplishment. Even though I didn't really get anything for beating it, I was really glad I played it till the end.
A bit more recent moment was partnering up with a co-worker to tackle Wesker in Resident Evil 5. We stayed up to 3am emptying every weapon we had in order to kill him only to find out all we had to do was run up to him and do a couple of button pressing cutscenes. (facepalm) The game is definitely more enjoyable playing alongside another person you know.