What you have described is more of a (good!) framing device for a story than a game. For the gameplay, you could take some inspiration from Alter Ego (
http://www.playalterego.com/). It does the life-in-a-series-of-episodes thing very well, though it's basically a text adventure (nothing wrong with that). I get the feeling you have a more restrictive story in mind than sandboxy games such as this, however.
As for games in which you can influence the storyline in a significant way: many people consider this to be a central goal of western RPGS. Many of these contain multiple endings (e.g. recently Vampire:tMB, Mask of the Betrayer, Fallout:NV), but these usually result from only a few choices at the end of the game. Some jRPGs and adventure games offer multiple endings too (e.g. Chrono Trigger, SMT:Nocturne, Vagrant Story), as well as games from other genres (Deus Ex, Blade Runner, other old adventure games). There are very few games in which your actions can change the whole storyline. A good recent example is The Witcher 2--the whole second act changes based on decisions you make in the first.
Note that the examples here are only from the top of my head and may not be the best representatives of this storytelling mechanic. If I left your favorite game out, it's likely not because I think it has less narrative branching. Anyway, post your own lists!