One of the things that have always endeared me to adventure games above all other types of fiction (books, movies, etc.) is that they give the player the opportunity to shape the story and unfold it at their own pace. While some games are content to have a linear story (and no slight against that — some absolute classics have only one straight solution), I am truly fascinated by the games that play up the "interactive" part of the medium.
While games like Indiana Jones and the Fate of Atlantis and Westwood's Blade Runner games did a bang-up job of giving us ample replayability value, I feel nothing comes close to the sheer mind-bogglingly malleable story of Tex Murphy: The Pandora Directive. How they managed to cram all that game content onto "just" 6 CDs is beyond me.
And what I truly love about it is that it's not just a case of "pick your path," like in Fate of Atlantis, but that the game keeps track of how you respond to NPCs and shapes the story accordingly. If you're kind and generous to people, you get put on the good path. If you're an opportunistic dick, you get sent on the bad path. And if you wibble-wobble between the two, you get sent on the middle-road path. And each path has multiple endings of its own!
What are some of your favorite games that let you experience the story in multiple ways?
Another one I really enjoy is Overboard, which is maybe stretching the definition a little but does fit under the detective and interactive fiction genres so I'll bring it up. I just loved how each playthrough takes about 30 minutes and the game has some systems behind the scenes to determine the ending you get so it's more than just Good/Bad ending based on one action. And if you screw up, it's not a big deal because the playthroughs are so short and you learn something in the process.