I have a couple on my list actually.
Shadow of the Colossus is up there. The Melancholy surrounding the game... wow, just wow.
Final Fantasy 8. Loved it. Loved Squall. It's very rare that you get to play a genuinely cynical, uncaring emotionally dead meat puppet, rather than some some pretentious jerk who is full of himself. I liked him because he just didn't CARE. The whole Squall/Rinoa thing was excellent, the Quistis subtext was great, all of it.
Kingdom Hearts 1 and 2, but mostly 2. Don't ask me why.
KOTOR. Mostly 1 for the twist (first time I ever leaped out of my chair and exclaimed at the top of my voice "What the merry Motherf*CK!?" over a videogame), and 2 because I liked the whole good/evil thing with Kreia. But I did feel a powerful empathy for Revan and the Exile (Revan moreso).
I'm going to say something controversial...
I didn't much like the ending of Fallout 3. It was just kinda... meh. Not a bad game, by any means. It was fun to play. but the ending? Meh.
GTA 4 in the context that it brought some personality and emotion to the series. I really liked the relationships between the characters and really got to feel for Niko. Specially with what happened to Roman, and the whole relationship (especially early on) between Niko and Dmitri. The bits and pieces about the "war" and the character's past, as well as the emails from Niko's mother added real depth. Really solid.