I've always wanted to get my hands on some golden oldies like Planescape: Torment. A lot of older games seem to have really well-written, fleshy, compelling stories. I guess you couldn't rely on flashy graphics and a physics engine to move copies way back when, so the story and the immersion factor were always large selling points. Ico is another gem I never got around to picking up, and have yet to try to this day. Grim Fandango too.
Though I'm certainly not saying there aren't any games with great stories being released today. As many have said, Bioshock is an excellent example. Bad Company, while not entirely original, looks like a bit of a laugh. I haven't played it yet either.
While we're on this topic, I've seen a few people mentioning Legend of Dragoon lately. I don't recall it ever being that popular, and I remember when it came out that only one of my friends had heard of it. I guess people were too into Final Fantasy to pay it any attention. But LoD had quite a good little story behind it, at least I thought so at the time. I've not played it in many years though, so I guess there's a chance that nostalgia is doing it a favour.