Must Owns? Well when I make must owns I always consider exclusivity on consoles, but there are to be a few exceptions.
1) Call of Duty 2, and Call of Duty 4 - I would skip 3. COD2 and COD4 both have insanely good Single Player campaigns which are both unfortunately very short, but COD4 makes up for it with it's pure ownage multiplayer. COD2 is more exclusive then COD4 obviously though.
2) Viva Pinata - Whacky and fun, It's more then just a kids game. You'll never have so much fun with inbreeding. Highly recommended.
3) Earth Defense Force 2017 - You'll probably be thinking I'm joking, but out of all the budget titles out there this is perhaps the most "fun", just remember the graphics are PS2ish.
4) Elder Scrolls IV: Oblivion - If the 360 was your only choice and you didn't have a PS3 or a beefy PC, then this is a must have for the 360. Between me and my wife we put in over 240+ hours into this game. Best FPS RPG that I've ever played, keep an eye on Fallout 3 since It's using the same engine.
5) Gears of War - Hands down the best third person multiplayer game ever... till GoW2 comes out.
6) Halo 3 - Despite it not being the best game of all time as Bungie wanted to hype it, It's probably the best multiplayer FPS out there when it comes to variety and fun. I would even rank it better then Call of Duty 4, but that's just me. The single player is good enough to warrant a play through, and the co-op over Live is pretty damn fun.
7) Ninja Gaiden 2 - You'll become best friends with death.
8) PGR4 - You can find PGR4 pretty cheap now since it was on clearance through Target. Don't bother with PGR3 despite it being even cheaper, because Bikes make this game a lot more fun and the multiplayer is more polished. This is the best racer IMHO on the 360.
9) Rainbow Six Vegas 1+2 - RSV1 was great single player and multiplayer wise. Part 2 stepped up the multiplayer although the single player was weak. Keep in mind however this is also on the PS3 and PC.
10) Virtua Fighter 5 - The best 3d fighting option right now, and while on the PS3 it did not feature the online multiplayer. Soul Calibur 4 might best this but that's yet to be seen.
Honorable Mention - Dead Rising.
Best XBLA titles - Contra, Super C, UNO, Geometry Wars Galaxies, Carcassrone, Hexic HD, Ikaruga and Soul Calibur.