I never owned a PlayStation One because my parents got me a Nintendo 64 at the time, but later on I played a bunch of PS1 games thanks to the backwards compatibility of the classic PlayStation 2.
I'd have to say that at the top of my list is Metal Gear Solid. That game is a masterpiece and a classic among late 90's/early 2000's gamers. It's honestly one of the most perfect games every developed, in my opinion.
On my list are other obvious hits such a Final Fantasy VII, Tekken 3, the Crash Bandicoot series, Spyro, MediEvil, Tony Hawk's Pro Skater 2, Resident Evil, and...I could honestly go on forever. The PlayStation One was a fantastic console.