Hrmmm, well numerous games have gotten me at one time or another for a bit, but very few have caused a deep degree of association.
I guess I could say that when I was younger the "Ultima" games, especially 6 and 7 (as a whole) did it to some extent. The fact that the hero is in some ways supposed to be you (well if you found a mystical portal in your back yard or whatever), and a lot of it involves striving to be the hero the people need, with the entire series being based around the concept of virtue to one extent or another. I got into it for a while.
One bit I still remember was in Ultima 7 Part II, where your basically visiting one of the other 3 lands (there were 4) from the first Ultima game, the continents having been seperated. This one was "The Lands Of Danger and Despair" where Shamino was king, Shamino having been a big player/companion since like Ultima 4, though he was present in the other games. After you cross a certain swamp and find out what happened to his kingdom in his absence while he was in Brittania, and that there was nothing you could do about it... well that got me a bit at the time, since there really wasn't anyone specific to smite with virtuous vengeance. The fact that Dupre also dies (which really wasn't nessicary IMO) kind of made Ultima 7 pt. II stick in my mind for a while, since these were just the kinds of things that didn't happen in most of the games I played, especially when I was supposed to be some kind of universe-saving super Paladin that always won. Of course then Ultima 8 and Ascension kind of blew chips after the highs and lows of the "7" Saga as Richard Garriot became obssessed with turning everything into a real time action game, and wouldn't even let me have companions anymore (kicking along with Iolo and company was half the fun), not to mention the fact that his dog must have died or something because he decided to make everything unrelentingly grim (at least in 8) and didn't even do it very well, kind of missing the point that half the fun of that kind of stuff is to purge the darkness. If the game wasn't crud enough about the time I started having to use Necromacy to survive as a major plot point was when I started to think Richard was losing it.... still I hope his return to game design will see some stuff similar to his earlier work but with new technology, ushering in a new era of RPG gaming.