5 games is sooooo not enough! There are so many great games for the 360, it is extremely difficult to limit my decision to just 5. First I had to go through a lengthy thought-process just to limit to these, but I will do it in sections so that it is easier to read. My shortlisted:-
Racing Games (In descending order)
3. Grid: Very nice racing game with some nifty features, such as the rewind time ability which allows you to rewind time back to the point before an accident, thus removing the frustration of having to suffer the rest of the race with a damaged car.
2. Midnight Club: Los Angeles: Very addictive racer with a nice combination of both motorbikes and cars, with nice customisation abilities, and a nice arcadey feel, there's nothing quite like flying through the LA sewer system at 230MPH.
1. Forza Motorsport 2: Tet's recommendation seconded, this is a great simulation game that has a lot of depth for both a purist racer and a casual player, although I must be honest and say the penalty system really annoys me, I mean when you go offroad, you're punished enough by the slowing down anyway, no need to penalize you for it, and it gets grating when another racer crashes into the back of you and you get a penalty for it. Nevertheless, for racers, this is the best offering out there on the 360.
May I also take a minute to mention RACE Pro, an ugly yet surprisingly addictive racer as the engine that runs it is really smooth and has a great feel to it, you can really pick it up and play, and if you can get over the unpolished graphics, this really is great fun to play.
RPGs
3. Lost Odyssey: Lovely Graphics, great gameplay and enjoyable all round, also with the heat system, there is an added intensity to the gameplay to make sure you don't run out of heat which helps make this a great game to play.
2. Fable 2: Very much for your casual RPGer, the first word that springs to mind when playing it is "Lovely" it just has a very homely feel to it, it's truly miles away from the grittiness of Fallout 3, but it is no less enjoyable and for me, as a casual RPGer, I found this a great play. My only criticism would be the lack of character customization, but apart from that, highly recommended.
(Also if you are into this genre of RPGs, but are looking for something a tad more mature, Oblivion is a very very very good alternative with lots to do, lots of customization and a lot of gameplay for your buck)
1. Mass Effect: Again following in the recommendations of others here, this is a truly amazing game, visually stunning, epic storyline, and truly vast aside from the main story, there are hundreds of things to be side-tracked (in a good way) by, and the overall feel of it is just great. Again, my one criticism would be that if you're a person who plays games with autosave too often and you forget to save, you'll find yourself really annoyed when you die and haven't saved in a couple of hours or the autosave in the game hasn't saved in hours either.
Action Adventure.
3. Crackdown: Very, very enjoyable game, being a superhero cop in a city is an exciting prospect, graphics (especially for explosions) are lovely and the gradual upgrades are a nice feature too.
2. Assassin's Creed: A brilliantly creative storyline, aside from the fact that the gameplay can sometimes feel a bit limited, and the fact that to escape from the guards you can simply sit on a bench once out of sight, this is a very enjoyable game, the swordplay is great and the ease with which you can jump from building to building and scale walls is fantastic.
1. Grand Theft Auto 4: Surprised this hasn't been mentioned yet, yes it is true that this is somewhat radical in that it ventures quite far from its arcadey roots, but the storyline, the voice acting, everything makes up for what is for a some a disappointment. There is no one part of this game that is boring, the aiming system has been further improved, my only criticism would be that the handling of the cars isn't great. Still, brilliant game, and furthermore, with the addition of the DLC: The Lost and the Damned (which is pretty much an entirely new game in itself) there's at least 50 hours of play out of this badboy.
(Also if you're into GTA style games, take a gander at Saints Row 2, which is also a very enjoyable game, but takes itself a lot less seriously than GTA)
Shooters.
Firstly, I will leave Halo 3 and Gears from this shortlist as you already have it
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3. Far Cry 2: Wonderfully vast, with so many things to see and do, it's the freedom of choice that really makes this game what it is. Do I run in with my grenade launcher and tear shit up? Or do I sleep in my local safehouse until midnight and then sneek in, picking off each man with my Dart Rifle and Silenced Pistol? Great game, aside from a not too great storyline, with lots of stuff to do, also very nice looking, and the map maker is great fun.
2. Call of Duty: Modern Warfare: Hugely popular because of it's online play, this is a must for those who are regular LIVE users, the progression and speed of play and really the sheer addictiveness of it is what has made it what it is today. Aside from the online play, the story is fantastic, if a tad short, and the post-Chernobyl mission in which you sneek through irradiated country has to be seen to be believed.
(Also if you like this, check out Rainbow Six Vegas, great story and great online play, just beaten out of this shortlist by Far Cry 2 as it has slightly more freedom of choice, but if you enjoy CoD, check this out)
1. Bioshock: Truly awesome game in every aspect, no other shooter for me, comes close. Visually stunning, with some of the best visuals you're likely to see on the 360 and a story to match, this game truly epitomises what is needed to be a great shooter, and the splicing adds an element of fun into the mix (Should I pick up this chair using telekinesis and bash the gene splicer over the head with it, or should I just burn him?) and with the ability to turn sentries into allies, no fan of shooters will be left dissappointed after playing this game.
Sports Games.
For me there are only 2 really worth mentioning:
2. Tiger Woods PGA Tour '09: If you're a golfing enthusiast, this is the best of the bunch, good visuals, enjoyable gameplay, and nice progression from newbie to champ.
1. FIFA '09: The only football game worth mentioning since Pro Evo became awful, all the licenses, all the teams, all the game modes, FIFA is now the daddy when it comes to footy sims. Gameplay is fluid, insanely easy to pick up and play and even a newbie to footy sims can score some absolute screamers if they put their mind to it. If you enjoy football, Get. This. Game.
The Best of the Rest.
3. Mirror's Edge: Innovative, addictive gameplay, very simplistic yet extremely enjoyable, this game will leave anybody who buys it happy, although the gunplay in it is abominable, if I'm honest
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2. Guitar Hero: World Tour: For any musician, this game offers the chance to guitar, drum and sing your way to the top. Great game for having a few mates around the muck about on, difficulty scales well and the drums aren't bad either. Lots of big songs in it and plenty to download if you so desire, a great game.
1. Rainbow Six HAWX: A recent addition to my game's library, this game is a riot. Story is fantastic (although it is set 15 years in the future, which leaves you wondering "why are they flying current day jets 15 years in the future?") the flying is easy to pick up, huge variety of jets in the game, and Assistance OFF mode is great and let's you pull off some truly awesome, and absolutely impossible mid-air feats.
So there you have it! My (hopefully comprehensive enough for you) shortlist of the best 360 games out there!