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PERSONA 3 PORTABLE

I CRy EVERY TIM ;~;

Crisis Core Final Fantasy 7, Dissidia Final Fantasy, Kingdom Hearts Birth By Sleep.

edit: Ratchet and Clank, all of them.

Ys series

Dissidia Final Fantasy 1 + 2 , Yu Gi Oh Tag force 1-6 ( man this is DRUG ) , Monster Hunter 1-2-3 and Phantasy Star portable 2 .

Also there is the final fantasy type 0 , and its really greet but its only in japan and no english release =( .

Patapon 3 , MH1-3, Persona 3 , Ys Seven

Persona 3, Valkyria Chronicles 2 and 3, Star war The force unleashed, syphon filter...... I will add more if I remember

god of war, kingdom hearts, burnout, daxter

tekken 6 and DJMAX!

only games I need on my PSP

Persona 2 and Persona 3 :D

peace walker

Monster hunter : Freedom Unite and Patapon 2, and 3.

Hmm..

Zettai hero project: Its somewhat a different take from NIS to their regular Disgaea formula, its quite enjoyable if you're into rogue-likes (You lose, your items are gone) Not really as deep as disgaea, as you just play as one character.

Prinny Can I really be the hero? : If you like hard 2d-platformers.

Ys the oath in felghana: Pretty nice action RPG.

Super Robot Wars Z2 - 1 and 2.

Corpse Party. But it doesnt really fit your requirements :(

Edit: Wait, you're looking into the Persona series? Those are the definition of anime-ish >_>

Metal Gear Solid PW
Beter Than That Anime Queer Shat.

My favourite is Corpse Party, which sadly might not appeal to you. Was ****ing scary though.

Other titles I enjoyed are

Burnout Legends
Castlevania: The Dracula X Chronicles
Coded Arms
Killzone: Liberation
Phantom Brave: The Hermuda Triangle
Ratchet & Clank: Size Matters
Resistance: Retribution
Shinobido: Tales of the Ninja
Tony Hawk's Underground 2 Remix
Untold Legends: Brotherhood of the Blade

Resistance: Retribution
God of War Chains of Olympus
Medal of Honor Heroes
Call of Duty Roads to Victory
Brothers in Arms: D-Day
Wipeout Pulse
Tom Clancy's Rainbow Six: Vegas
Star Wars: Battlefront II
The Simpsons Game
Silent Hill: Origins
Silent Hill: Shattered Memories
Patapon 1,2,3
Need for Speed Most Wanted
GTA liberty,vice city stories
Lego Batman

DJ max portable 2 and God eater burst

Too bad you have a Vita and not a PSP. I would've told you to mod your PSP and get some homebrew and translation patches for imports (though I hear modding the Vita is coming on strong).

As for original question: I know not all of these are available on PSN, but a modded Vita can run ISOs, as I hear (assuming you own the physical UMD, of course). * denotes my "must haves" even from this list.

Monster Hunter Freedom Unite (Unite if Freedom 2 expanded, and is the largest MH to date)
Gods Eater Burst
Star Ocean First Departure
Star Ocean Second Evolution
Dissidia Duodecim Final Fantasy (Original Dissidia isn't necessary: 012 is the original+expansion)
Dungeon Explorer: Warriors of Ancient Arts
Untold Legends: Brotherhood of the Blade
Phantasy Star Portable
Phantasy Star Portable 2 *
Patapon (all iterations)
Coded Arms