If you haven't played it already, the original Halo is worth trying out. In all honesty, you can skip Halo 2. I found the single player campaign disappointing, and the only way you're going to get multiplayer to work anymore is if you play one-on-one split screen or somehow organize a LAN party.
Mercenaries: Playground of Destruction is a fun title. As the name implies, most of the game's focus in on blowing stuff up in a variety of ways in a (mostly) open world environment.
Urban Chaos: Riot Response is an overlooked gem of a shooter. The story's a joke, but the gameplay is pretty fun. You can use a riot shield as portable cover and as a blunt weapon, and there are a variety of optional challenges in each stage that unlock weapon and gear upgrades, increasing the replay factor.
Crimson Skies: High Road to Revenge is an enjoyable flight combat game. The graphics still look impressive, and gameplay is reminiscent of Rogue Squadron. There's all sorts of optional things you can do like racing or finding hidden areas, and you can upgrade your aircraft with the earnings you make.
I heartily recommend Brothers in Arms: Road to Hill 30 if you don't mind a World War II setting. It's one of the most historically accurate WWII shooters I've seen on a console. You command a squad of soldiers, and you issue orders to them in real time. Good leadership is essential, and it almost takes on a puzzle element sometimes as you figure out the best way to flank that MG-42 pinning you behind cover.
If you're even remotely a fan of Star Wars, KOTOR is a must. It is one of the best Star Wars games I have ever played, no doubt about it. Admittedly some of the gameplay mechanics feel a bit clunky nowadays if you compare it to recent titles like Mass Effect, but the story is phenomenal.
While we're on the subject of Star Wars, you might want to try Republic Commando as well. Yes it takes place in the prequel era, but hear me out on this one. References to the movies are vague at best, and it focuses more on presenting the Star Wars universe in a darker, grittier light through the eyes of a soldier. Gameplay is a combination of Brothers in Arms and Halo, With intuitive squad commands and pretty darn intelligent AI, both friendly and enemy.
Those are just a few that come to mind. Personally, I feel the Xbox had an excellent game library.