My all time favorite PSP game is Dissidia: Final Fantasy. I have it installed on my PSPgo and I have the UMD I play on my 3000. I have put 196hrs 40mins 50secs into it since it's release, and I still play it on a regular basis.
I have a battle record of 2,832-313 (2,025 of those were won using Cloud, 143 were lost using Cloud). I've dealt 26,617,208 points of damage, and taken 6,017,090 points of damage, and I clock in a total battle time of 61hrs 33mins 47secs.
Even with all that, I still haven't unlocked everything, still haven't maxed out all the characters, still haven't beaten Shade Impule with every character, still can't beat a lvl 120 Chaos etc. I could easily put another 50hrs into this game, before I would even begin to be bored with it.
You guys don't have to get into that kind of detail, but what are your favorite PSP games? Don't be ashamed of your selections either, everyone has their own tastes.