I really can't understand how someone can enjoy Parappa the Rapper. I mean, it's just a matter of pushing a sequence of buttons shown on the screen. It manages to be more boring than making combos on Tekken 3.
Oh, and I agreee, X2 kind of sucks. I could list a dozen of thing that's wrong with it.
Here is my personal fav list of PSX games:
Metal Gear Solid - I guess I don't need to say much.
Castlevania Symphony of the Night - Amazing stuff.
Winning Eleven 2002 - Well.. it feels so dated now with Winning Eleven7/Pro Evolution Soccer 2, but well, if you only have a PS1, that's the best soccer game you can get for it.
Jackie Chan Stuntmaster - This is the closest thing you will get to a progressive beat'em up. Still it's more than that, it's a very cool game which only crime is to be a little too easy. If you are a JC fan this is obligatory
Micro Machines V3 - One of the most played games at my home for sure. It's nice on single player mode, but it's just the most awesome multiplayer game ever.
Bloody Roar 2 - Much like WE2002, it feels dated now with Bloody Roar 3 and 4, but on an PS1, it's the best 3D fighter you can get. It piss on Tekken 3's face.
Destruction Derby 2 - Gran Turismo is too slow and NFS Porsche has very uninspired tracks (But it's cool on 4 players mode). I personally think that's the best racing game the PS1 has to offer. Only big shame it's that it lacks a 2 player mode.
Metal Slug X - It's so widely known by now...
Die Hard Trilogy - It's 3 games on one, and all games rule a lot. Stay away from the sequel though.
Driver - Very cool driving experience, though it gets a little to repetitive some times. Stay away from the sequel too, it's horrible.
And much like Akira, I also a sucker for shmups. so instead of just cluttering the above list with them, I'll list the shmups I know in order I think you should get them:
Thunder Force 5
Strikers 1945 II
Gekioh/Shienryu (Though the saturn version is better)
The Shooter: Space Shot
In The Hunt
The Game Paradise 2
Erm.. actually, from "Ray Crisis" below I think you should pass. Raycrisis is still enjoyable for some, but the rest is pretty much crap.