Well , my PSP died sometime ago ( R.I.P - The battery is almost useless now and half of the buttons are long gone )

Here lies a listing of games from their respective genres which had proved itself exceptional.

RPGs

Persona 3 Portable
Persona 3 Portable - whilst the graphics and general atmosphere is watered down compared to the original - the choice of playing as a girl which leads to a slightly different plot flow - as well as the gameplay value of which Persona in itself holds - has labelled this game as one of the best RPGs on the PSP.

Fate/Extra
If your simply watching this game from Youtube - it would certainly appear boring as it seems to be a typical rock-paper-scissors game of sorts. But if you actually got around to play it , it is unusually addictive with three different Servants which could be summoned.
The Servant with the highest base damage ( and recommended for beginners ) - Saber.
The Servant with an arsenal of skills in his command - Archer.
The Servant whom requires time to nurture - Caster.
Depending on the character you have chosen you may have to adapt yourself to their combat capabilities - not to mention that there are unreasonable bosses like Arcueid and Ryougi in the game.

Star Ocean Series ( Remake )
Probably one of the more addictive RPGs on the PSP.
Not to mention - the game takes quite a while to complete. Finishing the story is one thing but you may not have cleared other side events or optional events scattered across the map.

Disgaea Series ( Remake )
Well , if you already have played the Disgaea series on the console it's kinda pointless to get it for the PSP - unless your aiming for some DLC contents like me :X
Still - considering the amount of time it requires to grind a character to a respectable level - keeping Disgaea around your pocket is more convenient than keeping it at home...

Valkyrie Profile : Lenneth ( Remake )
Whilst it is a remake , the game itself being horribly linear ( The Bad Ends and Normal Ends are boring so you might as well go straight for the True End ) and the replay value of this game is low - the story in itself is worth the deal.
I wouldn't recommend this game if your looking for an RPG that lasts for quite a while but if your going for the story , Valkyrie Profile would prove to be an intriuging tale.

Final Fantasy Remakes
If your a fan of old school RPGs - the remakes of the FF series is a decent place to look into.
Needless to say , most of you ought to know what a FF game is like... or rather , how the older FF games were like.

Phantasy Star Portable
It's alright if you just got the latest one and not the first installment - since the second one had some major improvements from the first.
The customization that this game have to offer is one of the reasons why this game is so very addictive - though on the downside , Phantasy Star Portable still requires to player to play Multiplayer modes to obtain materials for S Rank upgrades.
Whilst it does have Action elements - this game have a stronger RPG factor as compared to it's action factor.

Action

Monster Hunter Series
Needless to say , the Monster Hunter series would be a decent buy.
Whilst the action does kick in slow and the player have to go through plenty of repetitive missions before they can finally start fighting some decent monsters - the patience is worth it as the challenge it offers is great and you'd probably keep this game for a quite a while if your good with Action games.

God Eater Burst
Whilst it lacks the diversity value of which Monster Hunter have to offer - it is a decent choice if a player enjoys playing Monster Hunter - but wants an easier version of it.
Generally , God Eater Burst is more RPG-ish compared to Monster Hunter and the fact that you can have AI allies in offline battles can make one's life a lot easier compared to Monster Hunter where all the hunts are done solely by yourself. ( Felines are of some help - but hey , their not exactly perfect )

The 3rd Birthday
Despite the short campaign and the miserable revival of the Parasite Eve plotline , the gameplay in itself has proved fairly addictive.
It's essentially a shooter with a lock on feature but the OverDrive system in this game and the diversity of weapons makes it better.
Replay value of this game is quite high... and I've actually finished the game 50 times to unlock the final video in the video theatre - the shower video. ( I can't help it , it's Aya we're talking about here... )

Dissidia Duodecim
I wouldn't really label Dissidia as a fighter - it's more of an Action game truth be told.
If you have the original Dissidia - Duodecim isn't really worth the buy since the new cast of characters doesn't really make up for the fact that the original characters are pretty much the same as ever.
Not to mention the game became slightly unbalanced due to the unique moves that the new characters bear.

Mecha

Gundam VS Gundam NEXT PLUS
Imagine a Mecha game with some 3D fighter elements.
That's exactly what GvG is.
GvG can prove quite addictive even if your not a Gundam fan owing to the odd 3D fighter / mecha system combination.
( My favourite units were the Wing Zero , Wing Zero Custom , Strike Freedom , God / Burning Gundam and probably the Nataku. Tried using the Tallgeese but it's not really my type. )

Gundam Assault Survive
Regardless of whatever the critics might say - I love this game.
Whilst I am pretty annoyed by how the developers had exempted Gundam Wing and G Gundam , the fact that there are like hundreds of units to use and unlock is simply... AWESOME!
At first , this game may prove a little slow-paced and boring since you have to use run-down units at first ( Z.A.K.U , GM , Ball... you get the idea ) but you would slowly unlock more powerful units ( V Gundam , Hi-Nu Gundam and some other Armor types )
Coincidentally , the Extra Missions Bosses in this game can be a challenge. ( I took them all down with the 00 Gundams )
There is also a Multiplayer feature as well - so you can pit against your friends or fulfill missions along with them.
( Side Note : Totally whooped my friend's arse with the Hi-Nu Gundam... it's overpowered. )

Strategy

Tactics Ogre and Final Fantasy Tactics
Just grub these two at once if your into Tactical games.
"nuff said.

Valkyria Chronicles Series
The Valkyria series may prove a bit disorienting to newer players due to it's intergration of a simple shooter system into that of a strategy game.
What makes it unique is that the flow of the battle relies solely not on the wits of the player but also from active particupation from the player.
Still , don't expect something too tactically involved from this game - wits are required but true indepth thinking is reserved for nutty games like Shogun.

Music

Hatsune Miku : Project Diva 2nd
Easy , Normal and Hard mode may be quite ... easy , but what about Extreme mode?
I must admit this game had refined my reflexes in a way of which fighters could never hope to achieve...

Patapon
If Hatsune Miku : Project Diva was bad enough... meet your maker.
The player have to be heavily dependant on the music of the game to progress and I must admit - I found myself stuck every once in a while.

Fighter

Fate/Unlimited Codes
Easily the best fighter on the PSP.
Whilst full-chain combos are almost impossible to execute , simpler combos is very much possible for most players and the fact that the developers don't pull the stops in developing the characters as accordingly to the storyline is a very respectable feat.
Personally , I was surprised when I've managed to overwrite Rider's Bellorophon with Saber's Excalibur and Gil's Enuma Elish , in turn , would overwrite Excalibur.
Not to mention some characters , like Assassin , is more technical whilst others like Rider is swift and requires plenty of combos to perform in an ideal manner.
Still , Unlimited Blade Work's a pain to pull off - not to mention it hardly lasts a minute :X
( Need a remake of this game , with Shirou UBW with the addition of Fate/Zero and Extra Servants! )

Stealth

Metal Gear Solid : Portable Ops and Metal Gear Solid : Peace Walker
Whilst Portable Ops is a bit annoying ( At first ) , once you got used to it the game's actually simple enough to play.
Though the fact that you have to switch from character to character and that each of them can only carry four items is a pain in the neck.
On the upside - you can play as unique characters such as EVA or Ocelot.

As for Peace Walker - it's a far more refined game. You can still choose which character you want to use for side-missions but there is no longer the need to swap between members in the game as each mission is essentially a solo mission unless your playing with friends.
Not to mention the perks , jokes and Monster Hunter missions has transcended this game to the Throne of Godly Games.
On the downside - the camera is a mite bit annoying at times but you'd soon overcome it if your used to Portable Ops.
( The Dating Missions can be pretty awkward )

Don't think I've missed anything... or perhaps I have.
Oh well , no matter I suppose :P