For me, Jet Set Radio Future. In fact, I got it several days before I got an Xbox. I remember just staring at the box many times during those several days. I wasn't even a kid then, I was a grown man already! I think I was moving between houses at that time so that threw the timing off everything.
More recently, Valkyria Chronicles, and the possibility of Yakuza 3 being localized, got me to get a PS3... that and backwards compatibility so I could catch up on some PS2 games I missed.