Thanks for the thread, Khuutra.
For me, it's Dragon Age: Origins (on PC. I haven't tried it on consoles...)
Why?
1.) I was really surprised at the depth of the main party members and villain of the game. I've always felt that even great video game stories and characters have never come close to those in really good books. Dragon Age gave me characters who you really couldn't get to know in just a few hours of playing. While most of them didn't jump off the screen initially in terms of visual design, I found myself eventually thinking about the characters in my every day life and even having a few dreams about them, the way I would when I'm immersed in a good book.
2.) While Dark blood-all-over-your-new-jerkin Fantasy is my least favorite kind of fantasy, not to mention the fact that it's already become a cliche to me along with most action hero badassery, I really got into Dragon Age's world. They left a lot of things open about the world. Big questions like "who is god" (or who are the gods,) "do Dragons make small talk," "which dwarf leader is less of a bastard?" and "how much of a flake is Leliana, really?" were left ambiguous throughout the story, letting me fill the world in with my own theories and beliefs as I learned about things. Which is another thing that happens in good books.
3.) The first time I ever played Baldur's Gate, it was like someone had taken the party-based combat from the RPGs I'd always loved, taken out all the abstractions, and turned it into a completely realistic simulation of people fighting -- instead of selecting things from menus, you gave orders and then watched your fighters carry them out in real time whenever you unpaused the game world. Dragon Age makes that experience completely fresh again, by translating the Bioware tactical combat system into a 3D world and getting rid of the weird D&D ruleset that made things often feel oddly unresponsive and made numbers more of a focus than visual cues. If I'm playing a video game with graphics aimed at realism, I don't want to spend all my time looking at stat sheets, I want to really see and feel what I'm doing in the context of the world. In Dragon age, you don't even pay attention to your actual HP number -- you just look at the half circle around each characters' portrait that represents their life bar.
I'd also like the give an honorable mention to Demon's Souls, because it was also a really great experience -- especially in terms of the melee gameplay mechanics and the way the character stats affected them, the monsters, and the multiplayer.