The game I first played when I got my Xbox was GTA IV Episodes from Liberty City.
I got my Xbox 360 in 2009 for Christmas, I got it with quite a few games, GTA IV ELC, Halo 3, MW2, and a few others that I don't remember. The first thing I did after setting up the Xbox was putting in the GTA IV ELC disc, I literally didn't give a fuck about the other games until I completed The Lost and Damned and The Ballad of Gay Tony 100% (killing all the seagulls, completing side missions, ambulance things, etc). Why? Because it was actually the first time I played GTA IV, and I kept watching many videos of it daily that I just wanted to play it, and then I got it. So yeah... GTA IV ELC was the first game I played in my Xbox.