I hate lists, but here's my list of suggested titles:

Racing

Wipeout Pure and Pulse

Shooters

Killzone Liberation

Syphon Filter Dark Mirror and Logan's Shadow

Socom Fire Team Bravo

Sandbox

GTA Liberty City and Vice City

RPG

Brave Story

Disgaea

Final Fantasy Tactics

Jeanne D'Arc

FF Crisis Core

Valkyrie Profile

Platformers

Daxter

Ratchet and Clank

Castlevania Dracula X

Stealth Action

Metal Gear Portable Ops

Action

God of War

Pursuit Force 1 and 2

Action RPG

Monster Hunter

Gurumin

Adventure

Sid Meier's Pirates

Strategy

Field Commander

Puzzle

Puzzle Quest

Lumines

Crush

Echochrome

Music

Beats

Other

Patapon

Loco Roco

These are all highly regarded games according to me, the gaming media, and the PSP community at large. There's enough there to keep you occupied for years. The PSP covers a ton of genres with a good mix of the familiar and the novel. And I probably missed a ton of others.

@deathcape

things PSP don't have:touch screen and long battery life.....thats all

ds more had core games than psp ????you are referring to assassin's creed or something ??
well i played a ds way before i played a PSP and loved it...i was planing on picking up one but then my friend introduced me to the PSP and he had gba games on it(1.50fw) and i liked it...i just got it for gba games(strange) lol...but then i saw tekken dr and GRAN TURISMO coming and had to get serious with PSP games..me buying a ds right now is making a step back(my opinion)
ds sells better because it is aimed at the kids (16-6yrs)...everyone knows that

Does it have Bushido Blade ?  That would definately earn it bonus points.

Racing

Mario Kart DS

Shooters

Call of Duty 4

Brothers in Arms DS

Metroid Prime Hunters

Sandbox

GTA Chinatown Wars (out by Xmas)

RPG

FF III

Disgaea (not out in the US yet)

Final Fantasy Tactics A2

FF IV

Chrono Trigger

Valkyrie Profile (coming soon)

The World Ends With You

Etrian Oddessey I/II

Dragon Quest Monsters: Joker

Platformers

Super Mario 64 DS

Ratchet and Clank

Castlevania Dawn of sorrow/ portait of ruin/ order of ecclecia (coming end this year)

New Super Mario Bros

Stealth Action

not on the DS...sorry ^_^

Action

Ninja Gaiden DS

Action RPG

Dragon Quest Monsters: Joker

Legend of Zelda PH

Adventure

Legend of Zelda :PH

Strategy

Advance Wars DS/Days of Ruin

Fire Emblem DS (end this year)

FF XII Revenant wings

Summon Nigh Twin Age 

Puzzle

Meteod

tetris DS

Puzzle Quest

tetris DS

Music

Guitar Hero on Tour

Elite Beat Agents

Uendan!

Daigasso Band Brothers

Electro Plankton

Other

too much >_<

edit it later

PSP is capable of inking to the PS Store and download PSP and PS1 games along with themes and videos. Also with the new movie store you can transfer movies from your PS3 to a memory stick if you have a PS3 handy. lol

@MrBubbles


yea,you can run any PS1 game once your PSP is running custom firmware

well when i got my psp. gaming was my first priority, then it went to movies, i bought nothing but UMD movies for months,

now its just music, i havent bought a psp game since 06

Yea, because PSP doesn't have any games...

People always think the PSP doens't have many good games because the software is so low, but if you compare both libraries on Gamerankings or anything, it will be very close. It pretty much comes down to your taste in games.

If you haven't played any of the games, then you really shouldn't judge them.

Seriously? Get this out of here and put it in the DS thread. We all know the DS has a ton of good games. No one's arguing that point. How exactly do you expect people to not call you a raging fanboy again?

Edit: And it doesn't really help your case when you list games twice or list stuff that isn't out so you can pad your post.

Edit2: And it really really doesn't help when you list games that aren't actually on the system you're talking about (Hint: check out the "Platformers" section)  And I know you said you'd edit it later. I'm just messing with you.