My choice is an awkward tie between Fallout 3, Mass Effect, and Bioshock.
Fallout 3 is a fantastic and massive game that feels like every person and place (with the exception of the generically named NPC's) you encounter has a story, and when I put my mind to it I could be immersed in Fallout's world for literally months. But if at any time I played another game I couldn't for the life of me get back into it, and I was forced to create a new character, which seriously impeded my progress and willingness to reach the bottom.
At the beginning of Mass Effect, it feels like you're being dipped into an infinite ocean of win, but once you get your bearings and explore some more, you realize that not every planet is as massive and thought out as the Citadel (in fact the uncharted planets feel downright procedurally generated). So it's not an ocean, but it's still an Olympic size swimming pool, or maybe a slightly shallow lake, with a fantastic story and the genuine feeling that your choices affect the events taking place. Plus I really like the end credits song. I don't know, it just felt like a great payoff in it's own right for beating the game, like listening to Still Alive was for Mirror's Edge.
And as for Bioshock, it has an atmosphere that tops both of the others. This is most definitely because it's a much more focused game, your progress is fairly linear and there aren't any side quests to mention. But because of this it allowed the developers to really pour into the game an amazing art deco creepy vibe and atmosphere that truly complements the story of the city of Rapture. The story itself is pretty awesome as well, but not too good for replay value. and the controls are a bit sluggish, not refined enough to satisfy the shooter in me and definitely not intricate and varied enough for the RPG fan in me, leaving both unsatisfied.
So in the end I can't really decide which one is the deepest game I've played. Mass Effect gives the best first impressions of depth, but ends up shallower than expected; Fallout 3 is deep but hard to fully submerse into; and Bioshock, while not really that big, is definitely the most polished (and thematically accurate) dive of the three.
Oh, and sorry for the massive amount of text.