GRAW.  Soon after that came oblivion and cod2

NBA 2k 7--came with the console
Dead Rising and Gears of War. Gears was clearly the superior choice. :)

Gears of War. I was rather surprised I held off from buying a new console for a year... I guess WoW was big then.

GRAW and it was a very impressive game graphically and also very fun.  Gears of War after that which was mind blowing and at that point in time, it was actually a different experience than everything else.  I was so hooked on the online.
 
Technically, it was Hexic but that's no fun.

Amped 3, Kameo, PDZ.

Perfect Dark Zero, Gears of War, Forza 2, Viva Pinata and Oblivion (most of these came with the console or I got them first day)

Perfect Dark Zero. I thought the game was fun at the time, but then I eventually picked up Call of Duty 2 when it came out and didn't play PDZ as much.

Halo 3 actually.

PGR3 just before launch

Kameo, COD2 and Perfect Dark Zero.

PGR at launch.

PGR 3. That game was awesome.

Gears of War for me.

COD 2 or Perfect Dark, can't remember.

Kameo and PDZ

Was gonna be Ghost Recon, but my dad bought the regular xbox copy when he bought it for me. Still never played it.
 
I ended up getting THPS8 a couple days later.

Dead Rising, SFIV and Halo 3 + ODST came bundled. I didn't buy an x360 until last christmas. it was a kind of  a "why not?" purchase. 

I got Halo Wars and Assassin's Creed free with the console and I haven't played either.  The first game I paid money for was Eternal Sonata and it was worth it.

" Halo 3 actually. "

perfect dark zero and project gotham racing 3... I remember going to EB games launch day and being told that i would not receive my pre-ordered console till january, so I canceled my pre-order and walked out of the mall pissed off, then I saw SKU's being loaded off of a truck, and brought into a Zellers " Canada's shitty target " I was like hey, that could be 360's. I am here anyways, why not go check. So I did, and man it felt fucking awesome to have the cashier take box cutters to the SKU and take out the stores first console. Purchased it right then and there. Lasted me 2 years till RROD, 6 other console failures and here i am today. Now i have 3 working 360's and 3 broken ones.... 

Lost Odyssey

Call of Duty 2 on day one, Perfect Dark Zero a week later, then King Kong (I loved it) , and Project Gotham Racing 3,

Oblivion. It was pretty much one of the main reasons I bought a 360  and now it is probably my favorite game to this day.  I actually never even had an orignal xbox i used to play the gamecube lol...

MUA / Forza 2 come with the console, but the first game I bought was Beautiful Katamari. 

PGR or Dead or Alive or something.  Maybe Kameo as well...never got very far in that game, didn't think it was too great playing a fairy and all.
 
It really didn't have an outstanding launch lineup.....and people only hated on PS3's launch titles.

Madden 07.

This is a tough one, considering my system came with six different games upon winning it off of Ebay a couple years ago (surprisingly, it still works to this day). For me, it was either Madden '08, THPS:Project 8 or the original Gears of War.

Mass Effect. It was only 20 or 30 bucks at the time, and I knew it was supposed to be good.

" Oblivion.  I didn't like it. "

"GTA4. "

"@Abyssfull said:

"GTA4. "

You bought a 360 just for the GTA IV DLC? "

Burnout Revenge

Dead Rising and Perfect Dark Zero

Perfect Dark Zero and GRAW

Gears of War. It's the reason why I bought a 360 (before that I was primarily a PC gamer).

Perfect Dark Zero was the first one I bought. It was the "main" launch game for the 360 so I had to get it.

Gears of War and Chromehounds.

Call of Duty 2. 
Was a fantastic introduction to the next-gen.

I think it was Battlefield 2: Modern Combat.

GTA:IV... I wish I didn't have to say that.

Slightly embaressing really, it was Eragon and Cars (both given to me)  - they were both pretty terrible games although i did get strangley attached to the Cars game

Lost Planet right when I bought it, and then Gears of War from GameStop before I went home.

I came into the next-gen era quite late, so when I got my 360 it was bundled with Gears of War and Turok. I believe that GoW was scanned first though.

Need For Speed: Carbon

First Games I owned for my Xbox were Army of Two and Battlefield Bad Company.