The PS1 is currently the bargain-bin leftover of the console world, where its outdated hardware belongs. For those of us who are interested in trying out the most oddball and esoteric gameplay ideas that designers could stuff into a disk, that's the best time for a console. You can get a decent PS1 for $30 with pads, so there's no barrier to entry for trying new things, and many of these games are $5 if you know where to look, and that's all if you shop in-store.
The PS1 had what might have been the most diverse library of all the game consoles. The list of bizarre games is endless, but I've picked out a few multiplayer titles that I still play the hell out of.
BallBlazer Champions: a common bargain-bin game, imagine a Kart game reformatted into 1-on-1 soccer match. The game makes some odd design decisions - controls are extremely inertial, making it very difficult to aim correctly, but the game compensates with a "snap" button that quickly allows you to quickly target the ball. The graphics are very nice for the PS1, and the various maps are incredibly inventive. The weapons are pretty much what you'd expect - chainguns, missiles, mines, and my fave, the "ball caller". The game can be very frustrating because of its sluggish control, but in the end is solid and very unique fun. Its not hard, and its cheap, so pick it up and play through the singleplayer game on an afternoon. Quick recommendation: you might want to take a look at the control window, the default controls are the stupid "hold this button to strafe" instead of L/R strafing. Plus, it has the LucasArts classic 64/Atari game included as an easteregg.
Grid Runner: I can't recommend this little gem enough, and I see it filling bargain-bins everywhere. The game is very, very odd and kinda ugly. Its a top-view 1-on-1 maze game. One player is "it" the other is an unnamed position I call "runner". The map is full of flags that can be switched to a player's colour, and the game ends when a set number of flags (depends on the map) are your colour. Only the "runner" may switch flags. Fundamentally, the game is tag - to stop being "it" you have to touch your opponent. To fight, you have a variety of weapons including a stun gun, homing slowdown missiles, a teleporter, mines, and a bridge-building action (instead of jump) - but all of these take energy, which must be hunted down on the map. The maps are bizarre, with demented systems of ice, teleporters, catapults, speedboosters, conveyor belts, and other fun features that used to be plentiful in games but have gone out of fashion in the gaming world.
Evil Zone: I wouldn't think that an extremely rigidly structured fighting game would be good... and in many ways it isn't. But it is very inventive, and I find it fun when I'm in an odd mood. The game was quite obviously designed for the PC, even though it never made it there - it only has two attack buttons (like a PC pad) and it uses no "rolls" that are the bane of keyboard gaming. All of the attacks are long-ranged fireballs in the like, using simple combinations of "forward+attack" or "forward+attack+attack". Usually the "attack+attack" moves are secondary, less useful moves, so players quickly learn to think carefully and avoid button mashing. The characters are awesome - the game is a veritable "Battle Of The Stereotypes": There's the school girl; the Ultraman look-alike; the creepy little girl; the big oaf - they're all developed out in their TV-episodic "story" modes. Another odd feature is that the game has tons of scripted movie-style moves, which is both a plus and a minus for the same reasons as in Final Fantasy (Oooooh, that was cool... for the first couple of times, but now its just long).
Armored Core: While this game gained a lot of momentum on the PS2 in its later sequels, most of the players agree that the original PS1 title was the best in the series. The game feels like a smooth cross between Virtual On and Heavy Gear II. The robots are completely modular, and the parts all look spectacular and fit together nicely. Since everything in the game has a jetpack, the gameplay is bewilderingly fast. To compensate, the game uses a lock-on targetting system you'd expect to see in a jetfighter game. A must-have for giant-robot fans.
Carnage Heart: You won't find this one - its kinda rare, for good reasons. Robot design and programming as a video game. Like armored core, you buy parts and build robots, but a team this time. Then you program them. An interesting feature: different processors have different clockspeeds and memory spaces available for your program. The programming is a cute 2d grid based graphical language.
Destrega: an isometric fighting game with a wonderfully complex fireball spell system. You have an energy bar with enough power for 3 shots that recuperates quickly. Each button represents one attack type, Tidu, Este, or Fo - Fast, Strong, or Spray. By hitting the other buttons rapidly, you can burn energy to "modify" the core attack. So hitting "Este" then "Fo" fires a spray of "power" attacks, alternetly hitting "Fo" and "Este" takes your regular spray attack and makes it bigger and the projectiles more difficult to deflect. Combine that with a solid system of dodging, dashing, shielding, and other tricks makes a unique game. It also has one of the most amusingly bad "sexy" characters, who when "breathing heavily" looks like her breasts are pointing up and down.
Unholy War: What the Star Control team did for the Playstation. Essentially, it's like Star Control 1 or Archon 2 (two boardgame/action hyrbids from the same dev team) except with modern graphics and gameplay, using the "Mario 64" style of 3d platformer control. 14 different units to play, each with completely different fighting styles in-combat and special powers at the game's strategic map.
Future Cop LAPD: An unremarkable top-view maze shooter with a remarkable multiplayer deathmatch-strategy mode. In multiplay, the map is full of neutral turrets and bases that you have to "tag" to claim... as you tag more and destroy more enemy turrets and structures, you gain money to buy support vehicles and whatnot. Kind of a "1-click-strategy" approach. It gets old pretty fast, but its an exciting and interesting way to make super-simple action-RTS.
Critical Depth: Twisted Metal's long-forgotten little sister. This game is friggin' awesome. The catch is that it is ludicrously complicated. The control scheme is obscene, with each command button mapping to 2 different controls (doubletap the quickturn button to strafe, shield to deflector, brake to reverse, and thrust to afterburner), Twisted Metal style "special moves", an energy bar, a weapon inventory, a variety of vehicles to choose from, and an incredibly intricate and fun objective. The principle is this: you pilot subs around in a 3d underwater environment. You and a hodge-podge of other submersible treasure hunters are tracking down "powerpods" - big glowy spheres on the map. There are 5 on each map, and they appear on radar. If they are in a player's posession, then that fact is displayed on radar as well. Each powerpod gives you a QuakeRune-like ability (damage, speed, resist, etc) - which almost makes up for the fact that each powerpod you have also makes you more of a target. Once you gether all 5, you go to a portal to exit the level. This game has so much potential - it belongs on the internet, not trapped with only 2 players and a horde of bots like the PS1 provides. The soundtrack is an excellent epic synth/orchestral mix that is obviously of naval and pirate movies. In general, it makes up for its bewildering controls and stat-bar oriented combat with its wonderful power-pod based objective.
Brigandine Grand Edition: Pokemon for adults, really. The game pretended to be an RPG, but was actually a stunning strategy game. The game takes place on a warlords-style map, with each city producing "monsters". Unfortunately, the monsters need to be commanded by a hero, so you train and level up your heroes, each hero commanding a squad of monsters into combat. Battle is turn-based and takes place on an expansive hex-grid. In combat, the squads of monsters are each depending on their respective heroes, so there is tons of strategy surrounding protecting and attacking the enemy heroes. An excellent game - it looks like Final Fantasy, sounds like Pokemon, and plays like Warhammer. Unfortunately, only the limited release Grand Edition is multiplayer.
Bushido Blade: A fairly popular game in its time, but forgotten now. Bushido Blade took a completely orthogonal approach to fighting games - a realistic one. It focusses on swordsmen with a variety of weapons and styles, and unlike most games, takes their weaponry seriously. Anything more than a glancing blow in this game is a lethal one. As a result, the game is played round-after-round of extremely intense action, as one mistake is all it takes to lose. This is one that got a lot of use at parties over drinks.