Please recommend me a good interactive story experience

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Thank you all for the suggestions. Will check out To the moon, Another Code R, and I am already on Choice of games.

Will save Fate/Stay night for later, it seems like the kind of thing I would want to do in one go over a weekend. Thanks for the offer to help me install, maybe in a few weeks time. There is an anime of the same name, which is better?

I have tried fallout 1 and 2 before, they are not particularly easy games to be honest, reminds me of jagged alliance 2, in that you have the freedom to run into ridiculously difficult combat situations. I just died too much.

For Deus Ex, the graphics really haven't stood up so well because they went for a realistic approach instead of artistic, having recently played Human revolution (which was very good), I can't really look past the poor visuals.

I just looked up Catherine and Crono cross, they are interesting, so maybe laters.

If you're willing to persevere with Deus Ex, there's an excellent guide to installing graphics mods for it here: http://forums.steampowered.com/forums/showthread.php?t=1235278
Of course the game will still look aged as hell, but these will give you much nicer lighting effects, models, etc.

Failing that, my choice would be Red Dead Redemption. It has well drawn characters, terrific, exciting music, and a beast of an ending. I know you said you don't need graphics, but it's a very pretty game.