I'm more of a PC gamer, so don't have many 360 games - I have 4. As I have so few games I don't really have a clear favourite. The games I do have are Assassins Creed, Saints Row, Halo 3 and Rainbow 6 Vegas.
Assassins Creed, believe it or not, was the main reason I got the 360. I was interested in the "free running" aspect shown in the trailers and adverts - and when I got the game I was not disappointed. My only problem with this game is the cut-scenes and how you can't skip them. That reason, and that reason alone is why I no longer play the game.
Saints Row, even though regarded as many as a GTA "spoof" is one of my favourite games. I spent hours completing the mini-activities that have been provided - my favourite of which was the Hitman one. Once I completed all of these and got relatively far into the game, I found it hard to go on. Whether this was due to repetitiveness of the game or amount of time i spent on my other games can't be said.
Halo 3, like I implied before, was relatively similar to the previous two in terms of principle. Once I completed the game and got the majority of the achievements (skulls etc) I found myself going on R6V more and more. Since then the only time I go on Halo is to go online with my friends - whether it be to complete a few missions cooperatively or have a few matches.
Rainbow 6 Vegas is without a doubt one of the most replayable games I have been on recently. I started out on "normal" and completed it. Doing a few of the missions with my friends online only made me want to do it more - to try out new weapons. This led to be trying to complete it on the "realistic" difficulty. When I got to the harder parts of the game I stopped playing - as this is when I got into Counter-Strike on the computer.
Again, feel free to post your thoughts about your favourite game using either this topic or a new one. If there is a topic designated to a specific game please use either that topic or this one and avoid starting yet another.