Well, I have five that I would put into the conversation:
Grand Theft Auto: Vice City
Kingdom Hearts
Guitar Hero II
Burnout 3: Takedown
Portal
The first two are the ones that got me to buy a PS2 and the third was one of the first games I bought for the 360. I had a ton of fun with Vice City and the setting was damn near perfect, but the storyline really lost me after you took over the mansion and the missions started focusing on the racing aspects.
Kingdom Hearts was a game that I thought would be mostly Disney (not a bad thing in my opinion), but while it used Disney properties as a backdrop, it created its own world and fiction that just drew me in. I think I'd rather play KH2 over and over, but KH1 set up the world and their is just something special about the first time.
I didnt really think much of GH2 when I bought it. It actually sat in my house for about a month before I played it, regretting my impulse purchase. Once I got into it though, I played it everyday for like 2 months. My wife once asked me if it was one of my favorite games and my knee-jerk reaction was to say no. However, upon further review, I have to say that it is. I have RB, RB2, GH3, GHWT, GH:M but I still find myself going back to GHII every now and then.
Crash Mode is the reason I loved Burnout 3: Takedown. The symphony of destruction made this the penultimate racing/driving game IMO. I thought the races in Revenge were better, but the puzzle crash mode was best in Takedown.
Portal: I dont really have to say much about this game. At it's core, it's a simple puzzler: use the portals to get from point A to point B. But the story wrapped around these puzzles is what make the game special. The guidance/taunting of GLaDOS both compels the player to continue as well as entertains in its own right. A short experience yes, but one that is memorable. And yes, the song.
But if I can only choose one, for me it has to be Kingdom Hearts. I was sucked into the fiction almost immediately. It helps that I'm a Disney-Phile to begin with, but Sora, Riku, and Kairi stood out even over Mickey, Donald, and Goofy. I anticipate each installment of the series now because of this game. While some would consider it "RPG-lite", it has gotten me to give a look at series like Persona, Final Fantasy, and Dragon Quest, something I would have never considered before playing KH. It's a top five all-time favorite of mine, so it definitely qualifies as best of the "aughts" IMO.
P.S. Guitar Hero II is an all-time top ten for me. Just short of KH.