Bully Scholarship Edition(for its story), Fallout 3, Half Life 2(and both of its episodes), Getting Up: Contents Under Pressure, GTA: Vice City, And the Halo series(I wanna go back to pissing myself at the sight of the flood)
Call of Duty 4- The whole game would be great to replay, but specifically, Ghillies in the Mist. Goddamn, that is a great level.
Bioshock- My first playthrough absolutely blew my mind. I would do anything to meet Andrew Ryan for the first time again.
Mass Effect- Getting sucked in to the universe for the first time was awesome.
Red Dead Redemption and GTA4- Experiencing the story for the first time was awesome.
There are plenty of other games, but these are the big ones.
I'd have to say Dragon Age. For the first time in years (since Morrowind) it was an RPG game that I quite literally, didn't want to put down until I'd finished it 2 or 3 times. Was playing it on a morning before going to work, was tired at work because I'd played until 1am, it was a fantastic few weeks while it was all so "new."
The Darkness. For me the best part of that game is the stellar story and the way it is told. I've re-played it, but it just doesn't have the same emotional impact. It is the game I always reccommend people pick up in bargain bins or second hand for cheap. Who would have thought such a short FPS could be so touching (then again Portal).
Also I was going to say Zelda: OoT, but when I started reading the thread and some one said Oblivion, I realized that starting from scratch on Morrowind would be 10X better than Oblivion and OoT combined. I was just playing Morrowind the other night and it almost is like a second home because of how familiar it is. It's wonderful and disappointing because the game world is so well designed and I couldn't change a thing about it, but I almost want a new game so that I can explore it again. I don't know how many times I've traveled the roads between Balmora and Caldera, and I want it to feel new and dangerous again.
Tales of Symphonia. The story was absolutely incredible, and I wish I could play through it again not knowing what's going to happen (I've tried... still a great game, but nowhere near as good as the first time). Beyond that, I wouldn't mind having a second first crack at Bioshock or Oblivion either.