Might I recommend Katamari Damacy: a silly Japanese game that defies attempts to insert into a genre, has simple and intuitive controls, has cute stylistic visuals, and possesses a soundtrack that you will never, ever, ever, EVER get out of your head. And you will love it for it.
Beyond Good And Evil, a game in which you save a planet of animal people from a conspiracy Involving mutant humans in armor and fish-like aliens. (better than it sounds)
Deus Ex: The Conspiracy is also a fun conspiracy-fighting game that provides the best videogame RPG upgrading system I've seen to date, and plenty of text logs to read if you're into that. Shamefully lacking in the animal people department, though.
If you're into horror games, Silent Hill 2 is apparently the best game in the genre, and the only one I've ever taken the time to play. Wouldn't recommend playing it without having a GameFAQs guide on hand, but even with one it provided a good story and a tense, scary atmosphere that didn't need to fall back on the "eight or so dudes go into a place and all but one dies horribly" plot of most horror movies.
Spider-Man 2 got great reviews for a fun web-sligning mechanic that really made the best use of open-world sandbox gameplay, even if the rest of the game was OK to mediocre.
A word of warning, though: Don't bother with final fantasy. I borrowed FFX (final fantasy ten) from a friend, and it remains the only game that I can't bring myself to finish. It's not horrible, but it's unintuitive and combat got boring after a while.