I got a 360 pretty late myself (early spring 2009) so I was able to get lots of stuff on the cheap as well. Amazon Marketplace and ebay are some of my favorite places. But I'll also buy from retail chains if they have a good sale for a good game. I snagged Gears of War 2 for $20. The 360 has been around for awhile so you`re going to be overwhelmed with all the cheap budget choices available.

If were going by exclusives, I recommend...
Fable II (admittedly it is on the easy and short side but there is a lot you can do outside of the main quest and it`s a pretty fun cartoony rpg)
Blue Dragon (if you like jrpgs)
Gears of War 2 (I`m still on Act 3 but so far so good)

I see that Crackdown is getting a lot of recommendations. That is one game I really need to get. It`s dirt cheap now too. I gotta go try the demo soon since I might get this one next. Game demos are great. I wish all retail games had demos just like the XBLA games.

L4D2 on 360?
If you can buy for PC, the advantages and countless.

Started mass effect the other day had alot of fun then I put in forza 3. This game is everything I expected and more. I'm building a supped up camero to sell on the marketplace. I wanna see what the markets like before I work on anything else. I love the tuning options

A ton of people are already recommending you the standard games so I'm just going to say you should check out the xbox live arcade section. There are some really really great classics and new gems to find for amazing prices.

Indeed, Castle Crashers and BRaid being two of my favourites.  Secret Of Monkey Island is awesome too and the list just goes on and on.

It's more than fine on 360, I played the first one and it was great.

@Fame: Mass Effect is one of the best games of the generation.