My vote is on Bioshock 1+2
I figured someone would say
BioShock, and it indeed has a good story.
But, allow me to gush over
Thief: The Dark Project s'more here. I've been in love with it for 14 years after all.
The story was actually written by Ken Levine, same dude who did
BioShock. He was trying to make it in Hollywood as a screenwriter before moving on to games; he apparently got into games because Looking Glass Studios thought to hire someone from Hollywood to write the story for their game.
Anyway, the story in
Thief is deeply involved, very mystical, it has an interesting steampunk setting (cool, reserved steampunk sort of like the movie
Brazil but in a medieval setting, not that stupid bullshit cosplayers are doing today), and holy shit I dunno how much I can say without completely ruining the story for anyone who wants to play it, but it's great.
But yeah, you're this dude who as a kid was a pickpocket, you try to nab money from a dude in a cloak who's part of the Keeper Order, he recognizes your skill and takes you in, trains you, you leave the Keepers, you use the skills to not just pickpocket but extensively rob men of high standing, and...man, I can't go further without spoilers, lol. But there are a lot of fascinating characters involved, there are the Hammerites, a religious order who worship the Builder and have a boner for making and wielding hammers, there's creepy undead shit and ghosts... I really can't go into the twists and turns of the story, but what starts out as, "I'm gonna steal things," turns into this big story that's again really involved and interactive, and it goes into some really dark territory.
Thief II: The Metal Age also has a fantastic story. There's a new Hammerite denomination known as the Mechanists, led by a really eccentric and prophetic guy, and they bring about the Metal Age with awesome stuff like steam-powered security robots, really making the steampunk fantasy even stronger than in the first game. Amazing stuff happens there as well; you'll just have to take my word for it.
:O:O:O:O:O:O:O:O:O:O:O:O:O:O:O:O:O:O:O
And the mission briefs and cutscenes are a treat to watch. The lead artist developed a really cool method for creating unique animation (which also occasionally involved silhouettes of actual people), and he wanted to do the movie adaptation for
A Scanner Darkly and it's bullshit the producers didn't go with him 'cos the faux-animated look of the actual movie sucks cojones.
Thief: The Dark Project - Intro(I'm convinced, by the way, that the intro for
Batman Beyond was inspired by this game's intro.)