Crisis Core: Final Fantasy VII is the prequel to that game I know you know. It ties in quite excellently with Final Fantasy VII as it ends right when VII begins. The backstory of how many events came into place is a nice extension of the FFVII extended universe, and the gameplay isn't half bad either. The gameplay takes a more action oriented approach, similar to a hack and slash, but not quite. While for me personally, the game took a fair bit of grinding, and grinding boils down to mashing X against things weaker than you, overall, it's a fun game that people who own a PSP should give a try. (Warning: It contains MASSIVE, spoilers for Final Fantasy VII).
The Final Fantasy Tactics series will always have it's place, along with Fire Emblem as some of the best Strategy RPG series of all time (OF ALL TIME). Since they are SRPG's, they play a lot differently from the main series games. While I liked War Of The Lions, I think that Tactics Advance is probably the best out of the three. Had me playing long enough to complete all 300 missions as well as the secret Judgemaster missions. Nothing gold can stay however, as Tactics A2 sort of kind of was a little bit bad compared to the shining examples given by the former games. It wasn't terrible, but it just didn't have the fun that the other two games had. It was one of those games where you just complete the main story as the rest isn't worth checking out. Not to say some may not enjoy it, it just felt like getting cake for dessert twice, and on the third day, getting a muffin.
Dissidia Final Fantasy is fanservice at it's best. 2 characters from each game (1 from XI and XII respectively) all together in a Final Fantasy fighting game. Damn it sounds awesome. Well hold on to your pants, because it kinda is. While the story mode is an immense timesink and is overall bland, the actual gameplay is one of the most solid fighters in my opinion. It's a tad complicated and takes a bit to get used to, but when you master it, it feels really intense. ANother game that all Final Fantasy PSP owning fans should pick up, or at least try the demo.