PS1:
X-Men: Mutant Academy 2 - A great 2D fighter
Driver 2 - The first "real" modern open world game. It has pretty fun mission design.
Oddworld: Abe's Exoddus - One of the best games of all time, in my book. A great story, timeless gameplay, and great art direction.
Need for Speed: High Stakes - Quite a good Arcade Racer, and hold up pretty well,
Metal Gear Solid - If you've never played this, you should play this.
Harry Potter 1-2 - These games are surprisingly good games. They are sort of linear action adventures that play a lot like Zelda64
Chrono Cross - Amazing JRPG, unique, interesting and one of the best soundtracks in gaming.
Um Jammer Lammy - Pretty good rhythm game, with a lot of personality
Crash Bash - Considered by many to be the last "Good" Crash game until the remasters of the original games on PS4. Inspired by Mario Party, it's a party game.
Legacy of Kain: Soul Reaver - Dark vampire Zelda64-like.
Tekken 3 - A 3D fighter, maybe dated by today's standards, as the series has continued until today.'
Matt Hoffman's Pro BMX - While not quite Tony's Hawk's Pro Skater, it's close enough to be fun for fans of those, if you just want more content.
Spider-Man 2: Enter Electro - While not as good as the first game, it is still more Spider-Man fun. Would recommend to fans of the first.
PS1 games that may be better purchased elsewhere.
Final Fantasy VII, VIII, IX - All amazing and worth playing. [They can also be found with better ports on modern systems]
Crash Bandicoot 1-3 - Quintessential 3D platformers on the PS1. [They can also be found with remakes on modern systems]
Crash Team Racing - The Mario Kart64 competitor. [It can also be found with a remake on modern systems]
Spyro 1-3 - The other quintessential 3D platforming series on PS1. [They can also be found with remakes on modern systems]
PaRappa the Rapper - The original modern Rhythm Game. Full of personality and charm. [Was re-released on PS4 with updates]
Spider-Man - The original best comic book game. Loads of cameos, love put into ever inch of the game. Just all around fun. [Also available on PC and Dreamcast]
Tomb Raider 1-4+Chronicles - An amazing series for pioneering 3D level/puzzle design. Controls are kinda poop in this day and age. [Can also be bought on PC, and would probably play better]
Lode Runner: The Legend Returns - An amazing rerelease of one of the best trap-em-up games ever made. [There is a free fan-made reimplementation of this on PC that is better in every way]
Lego Racers - A Classic kart racer with some unique features. [While there is a PC version, it takes some work to get working in the modern day, but is otherwise completely better than this version]
Lego Island 2 - I consider this a classic that ran under the radar. Lots of fun mini games, and story beats. [Also on PC with 3 extra levels]
Tony Hawk's Pro Skater 1-4 - Amazing extreme sports+arcade platformers and true video game classics. [The first 2 were just re-released on modern consoles, and 3-4 were also released on PS2]
PS2:
Amplitude - Classic Harmonix Rhythm game.
Burnout Dominator - Spin off to the best arcade racer of all time.
Champions of North/Champions Return to Arms - Diablo-likes with local multiplayer coop.
ChoroQ - A CarPG, aka an open world Car RPG. Bizarre and interesting, if that's your scoop. (Sequel to Road Trip)
Frequency - Another classic Harmonix Rhythm game.
FullMetal Alchemist 1, 2 - While not amazing, if you are a fan of the original show, each game is like an extra episode.
The Getaway / Black Monday - While even clunky back then, they are very interesting games, with unique ideas and well done stories.
Ghost in the Shell: Stand Alone Complex - Average 3D action game, but new GITS content, if you want it.
Gadget Racers - Sort of a mixture of Gran Turismo and a kart racer. Unique and charming.
Gitaroo-Man - One of the most interesting and charming Rhythm games of all time.
Klonoa 2 - A clut classic 2D platformer. With the rumors of a remake of the first game, it's possible you might want to wait and not pay inflated prices for this, if they also remake the second.
Lupin the Third - While clunky, it's humor and charm are carried through pretty well.
Ratchet: Deadlocked - Spin off to the Ratchet and Clank series, more of that. [Also on PS3, but I've heard that version is totally busted]
Ratchet and Clank: Size Matters - Originally a PSP game, the PS2 port feels like another PS2 Ratchet and Clank game, if you want more. [Also on PSP]
Road Trip - Same series as Penny Racers, Gadget Racers and ChoroQ, a CaRPG, or Car-RPG, open world RPG where you play as a car.
Secret Agent Clank - Another PSP game that made it's way to PS2, yet another Rachet and Clank game. [Also on PSP]
Smuggler's Run - A fun party car game about collecting items, and trying to keep them away from other players and AI cars.
SOCOM series - I never got into them, but classic tactical shooters.
Space Channel 5 Episodes 1 and 2 - While originally on the dreamcast, these ports include all you could want. Basically a boxed set [Also on Dreamcast]
Star Wars: Super Bombad Racing - A funny joke game, if you want to pull out a game just to say "yeah, this happened"
Stuntman - Made by the Driver team, a fun, but insanely difficult stunt driving game. Includes an arena you can make, and build your own stunts.
Syphon Filter Dark Mirror / Logan's Shadow - Both original on PSP, they are great stealth 3rd person cover shooters, and hold up pretty well.
Timesplitters - While a little simple, it, gameplay wise, is a successor to GoldenEye/Perfect Dark, same team.
Twisted Metal: Head On - Originally on PSP, its a sequel to Twisted Metal Black, and has content from a previous sequel to Twisted Metal Black that was canceled [Also on PSP with less content]
We <3 Katamari - The best game in the series, but has never been re-released or ported. A reason to own a PS2.
007: Everything or Nothing - Third person beat em up, cover based shooter with gadgets, coop and other fun things. [Also on GC/Xbox]
007: From Russia With Love - A sort of Spiritual sequel, gameplay wise, to Everything or Nothing. [Also on GC/Xbox]
Beyond good and Evil - Huge cult classic, mixes Zelda, Pokemon Snap and Splinter-Cell with it's own DNA [Also available on PS3, Xbox 360/Xbone(Via BC)/PC/GC]
BLACK - Shooter made by the Burnout team, crazy explosions, and set pieces, nonsense story. Still fun to play [Also on Xbox/Xbone(via BC)]
Burnout 3 - The best Arcade Racer of all time. [Also on Xbox/360(Via BC)]
Burnout Revenge - The sequel to the best Arcade Racer of all time [Also available on Xbox/360]
Dragon's Quest VIII - A classic JRPG, in the long running series.[Also available with worse graphics, but more content and quality of life improvements on 3DS]
Devil May Cry 1, 3 - The original character action games. [Also available on PS3/360/Xbone/PS4/PC]
Driver Parallel Lines - A fun Driver spin off, with 1 city but 2 time periods. Some nice gameplay choices and fun overall set up [Also on Xbox/Wii/PC]
Escape From Monkey Island - The final LucasArts Monkey Island game, it has mixed reception, but I love it. [Also available on PC]
Gitaroo-Man - One of the most interesting and charming Rhythm games of all time. [A version with 2 new songs, but worse graphics was released on the PSP]
Half-Life 1 - The classic, with extra content, a separate coop campaign, split screen versus, Keyboard and mouse support, redone graphics, unlocked frame rate.
Harry Potter 1-3 - Well made third person action adventure games. [Can also be bought on Xbox/GC]
The Hobbit - A fun 3D platformer/action adventure game based on the book. Also a great soundtrack [Also available on Xbox/GC/PC]
The Hulk - Alright licensed comic book game [also available on Xbox]
The Hulk Ultimate Destruction - One of the best Comic book games, not many games let you knock over skyscrapers in an open world. Also the precursor to Prototype. [Also on Xbox]
Jak series - One of the quintessential PS2 platformer series on PS2, combing great story telling and fun gameplay. [Also available on PS3 and terrible ports on PS4]
Jak X - The Racing spin off to the Jak series, combining elements of Burnout and Mario Kart. [Also available on PS4 with tech issues, but fixing a save erasing issue from PS2]
Killzone - While this game has some technical issues, and can be janky, the story, art direction and level design are quite good. [Also available with less tech issues on PS3]
Lego Island Xtreme Stunts - While not as good as Lego Island 1 or 2, it's still got some fun mini games. [Also on PC]
Lego Racers 2 - The sequel. I don't think it's as good as the first, but it's still fun in it's own right. [Also on PC]
The Matrix: Path of Neo - A Third person shooter beat em up. Rather simplistic, but quite satisfying. Clunky, and does have some weird choices. [Can also be bought on Xbox]
Mercenaries - A very unique open world game that is an interesting sandbox. You can call in air strikes, do things more stealthy, mix on foot and vehicle combat in a war zone. [Also on Xbox/Xbox One(Via BC)]
Metal Gear Solid 2 / 3 - Play these games. Do it. [Also available on PS3/360/Xbone(Via BC)]
Midway Arcade Treasures 3 - The best way to play Hydro Thunder, probably. [Also on Xbox]
Persona 3 PES - Persona is a classic JRPG, but there is no 'complete' version of it, as the PSP version added and took out a lot. With rumors swirling around about a potential remake/rerelease, I'd say to wait. [Also on PSP with a lot of new and removed content]
Ratchet and Clank series - Some of the quintessential 3D platformers of the PS2 era. [Also released in HD on PS3]
Sly Cooper series - Another one of the quintessential 3D platformer series of the PS2 era. Totally amazing and aged super well. [Also on PS3]
Spider-Man 1, 2 - These games are pretty good, though as they are movie tie ins, they don't feel as fleshed out as the PS1 games. [Also on Xbox]
No One Lives Forever - A classic FPS, think Austin Powers, but better. [While not technically legal, it's easy to find copies on PC that are way better]
The Punisher - One of the best comic book games out there, totally captures the insanity of it all. [Also available on PC and Xbox]
Timesplitters 2, Future Perfect - Continuing the legacy of GoldenEye and Perfect Dark, the team kept pushing FPS gameplay. [Also available on Xbox]
Tony Hawk's Pro Skater 3, 4 - Continuing the trend of the series, these games continued to improve gameplay, and are just fantastic [Also available on Xbox/PC/GC]
Tony Hawk's Underground 1, 2 - Trying to branch out and try new things, the series continued to be good. [Also available on Xbox/PC/GC]
X-Men: Next Dimension - A sort of spiritual sequel to the X-Men: Mutant Academy series, but taking some cues from Dead or Alive, it's not bad, but also suffers from pacing issues as well as technical issues. [Also on Xbox/GC]
Twisted Metal Black - Dark twisted, and unique, it's hard to explain how interesting this game is. [Also ported to PS4]
Those are the ones I can give comment on. There's a lot I didn't play/play enough of to really comment, so its not an exhaustive list. I mean, there's a lack of many big things, like Resident Evil, or Silent Hill.
