Hmmm, well from the way I'm reading your posts, you sound like a PC gamer. Honestly as a PC gamer myself, I think the 360 has probably the least amount of games it can offer due to how many games they share. A PS3 is usually a better pick up due to the exclusives you cannot get on PC or 360. That doesn't mean it still doesn't have a lot of worthwhile titles, but just a lack of exclusive titles. Especially since unless it's developed by a Microsoft studio, you never know if it will go to the PS3. Bioshock and Ninja Gaiden 2 come to mind.
With that said, Xbox Live Arcade is a definite strong point of the system. Loads of different titles to play and a free demo to try every game. Some of the strong titles are Shadow Complex and Castle Crashers as it's already been stated, but 'Splosion Man, the two Banjo games, soon to have the first Perfect Dark, Trials, and many others I can't think of right now.
Exclusive titles I can think of as must have is definitely Lost Odyssey. It may not have the labeling of a Final Fantasy game, but it sure as hell feels like one. A great classic JRPG. Gears of War is a good romp for single player Third Person shooting, but I hear the multi blows. Forza 3 is well loved by the racing fans. If you can find a cheap copy of Kameo, it's a nice short action game from Rare, but may be hard to find due to it being a launch title. Also look into the two Viva Pinata games (not Party Animals) if you want fun and creative Simulation games. I'm sure others will say Halo 3 and ODST, but I'm not a Halo fan.
Really the must have games on the console are ones that are not exclusive. Fighters would be as already recommended plus Dead or Alive 4, Blazblue (the spiritual successor of Guilty Gear), and Virtua Fighter 5 (best version of the best 3D fighter). Action oriented games such as Batman, Devil May Cry 4, Assassins Creed, and Ninja Gaiden 2. Stuff like Rock Band and it's sequels are must have.
Edited, Nov 14th 2009 11:32pm by OnimenoJason