Two amazing games I highly recommend are "Ico" and its spiritual predecessor, "Shadow Of The Colossus". The detail in these games is so fine and intricate you can tell that the developers put so much effort into it. They are works of art! Epic and at the same time emotional.
Lately I have played Rollercoaster tycoon 1&2 over and over again.. Those are very lovely, I guess second is even better. I don't usually play on computer (except those) but I remember when we have huge Amiga game partys with my friend when we were young.. I don't even remember what we play but we have so much fun!
I love play Crash bandicoot, especially second, third and CTR. Never learnt to play Crash on xbox. And of course Wipe3out, even it was little too hard for me such as Vib ribbon (I loved idea so much but never have to time learnt to play it). ooh.. maybe I should try to rescue Crash and his friends again?!
On xbox I like Fonza and Burnout 3, otherwise I think that xbox don't have many games which I like, well, expect Halo or something. One of those games I have ever passed (well, team play..)
Now I wait new Grand theft auto. I haven't play any of GTA but I have followed when my bro plays and I love it. Actually following or playing together with somebody is the best thing when you're playing, in my opinion.