Oh man, so many good games. SO. MANY. I don't know where to start.
Metal Gear Solid Integral - The best version of an amazing game. This has the VR Missions disc and a first person viewpoint as its main additions.
Tenchu Shinobi Gaisen - The best version of the original Tenchu. It's the American version with Japanese dialogue, plus a LEVEL EDITOR for your own custom made stages.
It was limited compared to the sequel, but so awesome at the time.
Tenchu 2 - This game had some issues (mainly due to the graphics engine), but it offered three characters with some different paths for each, plus an insanely awesome level editor that was far better than the original's. I spent hours making custom levels and downloading others', playing them on VGS.
Bust A Move Dance And Rhthym Action - I still play this game even today. Such an awesome dancing game with fun characters and music.
Bust A Move 2 Dance Tengoku Mix - Love this one just as much as the original. I wanted to see a sequel so badly, and when it finally did happen as Dance Summit 2001, it was a pile of crap. This was then followed up by BRITNEY'S FUCKING DANCE BEAT.
At least two amazing games came out of this series...
Street Fighter Alpha 3 - It's far from being the best version of this game, but in December 1998 when the Japanese version came out, this port was mind-blowing. Unlike previous Capcom PSX ports (except for the original Alpha), this game retained more of the animation, and Capcom just went NUTS adding in truckloads of additional content. There's the world tour mode, Pocketstation support, extra characters, stages, and more. In terms of extras, Alpha 3 (on any platform) is the best home version of an arcade game that Capcom has ever made.
Real Bout Special Dominated Mind - SNK never really did all that well on the PSX. Most of their ports were choppy, load time-filled messes, especially compared to the generally superior Saturn versions. But SNK did good with their Real Bout games. Even the original Real Bout was an excellent PSX port, cutting very little animation overall. But for RBS, SNK decided to change up the game, and what we got is basically an upgrade to RBS, and one that completely differs from Real Bout 2, which followed it. SNK removed the line-sway system, eliminating animation cutting issues, and then ADDED more background features and animation. Characters got new special moves, plus unlockable super moves. Super-cancelling was added. Alfred from RB2 was made into a fully playable character. A new end boss, White (who is clearly based on the main character from A Clockwork Orange) was added. Plus a new anime intro and anime cut scenes in single player were added. This port kicked ass.
Tobal 2 - This game still holds up today, thanks to its silky-smooth 60fps game play. Tobal 2 really improved on Tobal 1, and with the dungeon mode, tons of characters could be made playable. But it was the main game and character set that rocked, combined with the grappling system and the projectiles, that really made Tobal 2 shine. It is an absolute crime that this game never came out in English.
Castlevania Symphony Of The Night - The best Castlevania game ever made. That's all that needs to be said, really.
spoiler (click to show/hide)
It's also the best Metroid game ever made.
Parappa The Rapper - Fun rhythm game from Sony with some cute characters. Freestyling was fun too. I thought Parappa was going to blow up and be a major mascot for Playstation; at one point there was an anime series and even a restaurant in Tokyo (Cafe Rodney), but it just sort of faded away with the PS2. Parappa 2 was just a blip on the radar.
Umjammer Lammy - The follow-up to Parappa that contained multiplayer and an entire hidden Parappa mode, with all the rock songs done in hip-hop style. And THEN you could COMBINE the two together for Lammy and Parappa game play awesomeness.
R4 Ridge Racer Type 4 - Namco's final PSX racer had some awesome cars, tracks, and tunes, and a nice large single player mode. I liked this game more than Ridge Racer V. Actually because of R4, I was pretty disappointed with RRV.
Jumping Flash! - Awesome first-person platformer/shooter. IMO it is still the best 3D platformer of its time...better than Mario 64.
Jumping Flash! 2 - More of the same, but that's a good thing.
Ghost In The Shell - This might as well be called "Jumping Flash 3." Developer Exact followed up JF!1 and 2 with an anime licensed game. And surprisingly, it didn't suck! It was great! You control the Tachikoma tank and can walk and climb up or over any surface, shooting bad guys as you move throughout the levels. The game also featured anime cut scenes that looked far more appealing than the original Ghost In The Shell movie; the art looked like Shirow's designs. They also kept the same voice actors. And the soundtrack FUCKING ROCKED.
Street Fighter EX+Alpha - This game seems to get a lot of hate nowadays, but I don't remember that being the case back then. Anyway, SFEX was Capcom's forray into "3D" for Street Fighter. Only it wasn't. Arika took a cast of mostly original SF2 characters and put them into a game with a bunch of new, yet familar characters. The graphics may have been polygonal, but the game play was entirely 2D. A more old-school, Super Turbo-like play style was used, combined with super cancelling, allowing for multiple supers at once. For the home version, Arika added Sakura and Dhalsim (hence "Alpha" in the title), ending CG movies, and an awesome, awesome arranged soundtrack. EX2 rocked as well, but I have a preference for the character selection in the original. Arika would later go on to make Fighting Layer for Namco, a great follow-up to EX2 with better game play and an entirely original cast, but it never got a home port. And then they fucked it all up with the rushed EX3.
Viewpoint - Visually enhanced port of the Neo-Geo shooter from EA. Different soundtrack, but still good...despite some frame rate issues, I enjoyed this port.
Rival Schools United By Fate - Follows SFEX as being a more 2D like fighter with polygonal characters. This was easily Capcom's best new I.P. following Darkstalkers, with a quirky cast of high schoolers and their teachers, drawing from various school clubs and archetypes to make a fun versus-style fighter.
Resident Evil 2 - The follow-up to the original Alone In The Dark rip-off was a bigger, better game with four different scenarios to play through, and lots of violence, gore, n' puzzles to solve.
Resident Evil 3 - RE2 1.5 follows Jill in her escape from Racoon City. We also find that she moonlights as a hooker, since nothing else can explain her choice of attire.
Dino Crisis 2 - This is the best survival horror game on the PSX from Capcom, better than all the RE games, and far, far better than the boring original. The game is almost devoid of puzzles and instead focusing on tons of pulse-pounding dino-killing action. It's faster-paced and controls better than the typical "tank" RE-style controls as well. Still worth checking out today if you've never played it before.
Strider 2 - A bit on the easy side, but the follow-up to Strider kicked ass. The original was included as a bonus, too.
Einhander - The best game from Square not named "Tobal 2."
Slap Happy Rhythm Busters - This import-only cel-shaded 2D fighter had a very unique look and feel, and great game play. The developer's previous games were the incredibly shitty Vs. and Shaolin, but the third time was the charm for them. They scrapped their 3D crap and went 2D, adding a very stylish soundtrack and art style, plus rhythm game-like finishing moves. Another title that still holds up quite well even today.
MTV Music Generator - There's far better music creation programs out there, but at the time, this kicked so much ass as a console title. I loved making songs.
Raiden Project - When I was a kid, I loved Raiden in the arcade. LOVED IT. But then I moved and never saw the game in an arcade again. I forgot the name, but I remembed how awesome it was. Imagine my surprise when I first got my Playstation and found out that the original Raiden was that game...and it had a sequel! I never took well to Raiden III (BORING!), but I really enjoyed Raiden 1 and 2.
Motortoon Grand Prix 2 - This game came out back when Polyphony knew how to make FUN games, not boring sim garbage. A really great mix of racing and Mario Kart-style game play, and some nice bonus features. I enjoyed this game more than any other "kart" title that followed it.