I hope I can be forgiven for not having a Final Fantasy on my list.
1. Arc the Lad (1, 2, 3)
Since these were all released at the same time in the US in the same package, I'll put them all in one entry. The closest thing to Shining Force I've found in the US, and not as overly complex or elitist-spawning as Final Fantasy Tactics.
I can't explain why I like this game so much. The graphics are awful, and the controls are just a little bit stiff. Still, the fact that the 4 characters play as differently as they do, and the fact that it's the best 3rd person action shooter on the system make it one of my favorites.
3. Silent Bomber
A fast paced action game that felt like a cross between Smash TV, Bomberman (but not as slow), and some vertical shooter I can't quite put my finger on. The voice acting in the US version is amusingly awful, and the story isn't great (but in a game that's not an RPG, I don't place a premium on things like that), but it is incredibly fun and addictive, with a flexible system to allow you to be offensive or defensive minded depending on your preference.
4. ThunderForce V
Though the Saturn version was better, the PSX version does just fine in my opinion, with classic horizontal shooter gameplay interspersed with updated graphics. I don't like how Victor Ireland likes to screw with the difficulty settings when he translates a game for an American audience, but even that isn't enough to ruin a game that I believe has the best music ever for any PSX game (Valkyrie Profile is a close second)
5. Bloody Roar
I fully expect to be strung up by my toes and fed to HHH for this one. It's a Fighting Vipers clone with a hefty gimmick thrown on top of it, but I was entirely too taken in by the gimmick, and at the time it came out, it was a good alternative to the Tekken series.
Honorable Mentions ------------------ Gradius Gaiden Metal Gear Solid Valkyrie Profile R4 : Ridge Racer Type 4 Soukyu-Gurentai
I have to go with Final Fantasy 7 as my favorite.. It was the first game that wrapped me up in it's story so much that I just HAD to beat it. There is so much stuff to do that it never truely gets old. I still pop it in every once in a while to breed some Chocobos and play some games at the Golden Saucer.
I also love Fifa 97 and still play it religiously for some odd reason.
Power Move Pro Wrestling was the coolest way back in 1995. Anyone else still play that game?