I bought a PS2 to play Shadow of the Colossus. It's still one of my all-time favourites. I played it for a short bit on a friends PS2, and was basically thinking "I need to have this as soon as possible". So it was the game I picked up the PS2 with, and my expectations were met, then blown away.
I bought a 360 to play Bioshock when it came out. I actually bought Bioshock before I got the 360. I think I actually played Quake 4 and Gears first though, because I wanted to wait until I was back home to play Bioshock, to give it the right attention.