I'll give you a list of my personal favorites.
Lumines 1 & 2: very addictive. If you like Tetris or 2D puzzle games in general you will love it. The game play exactly the same for both except different skins and soundtrack. Its just a great game. Its a must have for anyone who owns a PSP. Sazzy plays it so much I think she beat me last time.
Metal Gear Solid: Portable Ops: Its just like playing the original MGS series. But whats unique about this is that you have recruit enemies and build your own rebellious army from the inside. Excellent storyline. The cut scenes are just like the MGS digital graphics novels. Has voices and sound effects and the art is amazing. Basically its like watching a manga come to life. Also the online play is great.
Marvel Ultimate Alliance: If you like Marvel characters you'll love it. It has the game play of the X-Men Legends series with 4 man teams and simple combos. Lots of fighting and good story.
Final Fantasy Tactics: War Of The Lions: I love this game. Very similar to the PS1 game but its different. I find it a bit difficult to learn techniques but the story has me hooked.
Valkyrie Profile Lenneth: I don't talk about it much but I love the game. A great RPG and the more you disobey the gods the better the ending you get. It is a remake of the PS1 game.
Metal Gear Ac!d 1 & 2: Most people won't like this style of gaming because you have to collect cards. Thats right your weapons and gear are cards. And its a turn base strategy game. I didn't think it was going to work but its great. And there is nothing better than seeing a Metal Gear in 3D. The second game was made for 3D game play which looks great but the black 3D box thing was stupid. Best to play it when your alone. Holding a box to your face looks dumb.
A few more great games are Dragon Ball Z Shin Budokai, Tekken Resurrection (which is the highest rated PSP game so far), Prince Of Persia: Rival Swords, Grand Theft Auto, Street Fighter Alpha 3 Max, etc.
A lot of games are just remakes with improvements but they are worth playing.