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Got a thing going on with Morrowind and the Nerevarine storyline.. Though, only a lil bit. :)
Bioshock was fantastic.

"A man chooses. A slave obeys."
Dawn of War
Neverwinter Nights 2
Been replaying Dragon Age -series. It's still immersive as hell, and not letting go. At all. Truly a great game (both of them). I can't wait for DA:I...
The first time I can remember getting truly immersed in a game's story was Final Fantasy XIII. I don't know why, exactly, but it was just amazing.

The last time it happened, however, was with the Mass Effect games. Having decisions carry over from game to game was brilliant, and I felt like I had a true part in the story (even if the outcome is roughly the same, regardless of player decisions).
The recent ones are The Cat Lady and The Walking Dead. Also, ME2 (then ME3 made me lose interest in the whole series because of too many immersion-breaking moments).
I tend to glaze over story unless it's core to the game play. I might vote for Alan Wake though. I actually paid attention to the story and cut scenes, which I don't normally do.
Halo CE (1), then Halo Reach, you could not remove my head from either even when i was at work! - god knows what the customers thought when i was chirping out with stuff and drawing tactical maps!

erm now there have been one or two others but trying to think which!
Red Alter (1) was my first game i put night and day into, then oh of course how could i forget! Star Trek Voyager EF, then was halo then erm the ones i cant remember and then reach a few years ago.

ive been really lucky with all them now i think about it!
Post edited September 23, 2013 by chezybezy
I felt deeply immersed in both Deus Ex games, The Cat Lady, Alan Wake, and many other games. In fact if they have a good story and I feel an urge to push on I often fall into the immersion, even when the main character is a complete ass.

The first time I really felt immersed into a game it was Dungeon Siege. After I saved Ulora from the Krug, then Zed joined me I felt like I was leading a party to save the land, and then the world... A shame that when it was all said and done I had to leave such a happy life behind.
I can't quite remember which the last game was that had a story that I completely immersed myself in, but I'd say The Secret World, which I started playing this summer, comes pretty close. The story is a bit too vaguely defined at times to follow, but the codex entries pull me in every time, and every so often there is a moment where you manage to connect two small, seemingly unrelated pieces of the story and suddenly it's one of the most intriguing settings in gaming.
Thomas Was Alone
The Swapper
39 Steps.
Witcher 2.
999: 9 Hours, 9 Persons, 9 Doors for the Nintendo DS.
System Shock 2. The main plot was certainly interesting, but I was really taken by the crewman vignettes: The panicked scientist trying to objectively describe the digestive tract in which he is trapped, the two lovers trying to find each other in the carnage, even the researcher wondering where her nursing staff has gone. I don't play the game any more, but I do listen to the audio logs as a kind of radio play.

Avernum. The world and the plight of individuals within were described well. I often found myself imagining it well after playing.

Gothic 1 and 2. Again, the world is really nicely done and the inhabitants are gritty and a grim and so unlike the flat characters I meet in other games.

Now that I reconsider, I don't think the central story in these games is as immersive and memorable as the small stories told within. I definitely fell into these games, but maybe I should attribute that to convincing world rather than terrific story. They're hard to separate, I guess.