Hi there. This is a wonderful question! I'm sure I'm missing some, but the ones that come to mind right away are:
Myst (I LOVED this game so much, but I also loved Rand Miller and wanted so much to be Catherine to his Atrus!) When I think now of wandering around that world, I do feel like I was really there and I never wanted to leave.
Oblivion: I am STILL playing this game more than four years after it came out (on and off) and in the early days of play I thought about it so much that I actually would dream of it, and even now I feel like I live there; like I met those characters, lived in those houses, killed those enemies, etc. Today while walking the dog I noticed some bushes in a nearby neighborhood and my first thought was, "Harrada bushes, in Colorado!" LOL!
Morrowind: I just recently started playing it again and since I never got far the first time, this really is like my first time in alot of ways. Already I can hear the voices and the music in my memory so clearly, and there are certain characters I dislike or like so much that they seem to be a part of my real life sometimes. And btw, I don't know who does the voices for the characters, but I love the male voice that often says, "You may speak". I feel like swooning to the floor when he says that. LOL
Darkness Within: I really loved this game and there was something about the settings, the light and shadow, the sounds that made me feel like I was really there in a certain house, or office. I felt like my own sanity was being tested, and that I was investigating this whole story for real. When I think about it now I wander around the rooms in my memory and really want to go back there for some reason. I guess I fell in love, what can I say?
I guess that's my short list for now, but I want to thank you guys for coming up with this topic. It's so fun to think about and to realize that I'm not the only one who's grown really attached to certain games. My husband really doesn't "get" video games or why I like them and frankly, none of my RL friends do either, and so it's fun to have a place to talk about stuff like this.