There might be (for sure) better games than this top 5 of mine, but these are the ones in which I've spent more hours:
Tekken 5 Dark Resurrection
OutRun 2006 Coast To Coast
Street Fighter Alpha 3 Max
Metal Slug XX
Final Fantasy Tactics War Of The Lions
For example... Tekken 6 is technically better than Tekken 5 DR but I played (and I still play) the PS3 version...
God Of War Ghost Of Sparta is considered for most people the best game of the system, but I didn't had the time to enjoy it...
MotorStorm Artic Edge is pretty fun, but out of this top because of OutRun or even Gran Turismo...
Any fun tennis sports game like Everybody's Tennis, Virtua Tennis 3 or Smash Court Tennis 3 could have entered my top 5...
Pro evolution Soccer 2014 too if you dislike mobile versions like me...
Lumines, Tetris or Puyo Puyo for an arcade puzzle... Maybe FF Crisis Core instead of FF Tactics...
Or Castlevania Dracula X Chronicles, MegaMan Maverick Hunter X, MegaMan Powered Up, even Metal Slug Anthology instead of Metal Slug XX...
Depends on the day and the weather...
PSP was a very good machine, I'd just change its screen to fit better with modern standards and keep enjoying the original system even today...

Fortunately, we got PPSSPP and the emulation is sooo great!
