I was a pretty stupid PS1 owner, as I mostly played arcade-games on it, while still playing snes, nes and megadrive/genesis-games on PC-emulators. I actually just started to seriously explore its library a few years ago, which means I have still only played its top games, however, here are a few decent ones:

-Metal Gear Solid - my favorite game so far for the system
-Final Fantasy VII - extremely overrated, but still a good game. Nothing compared to its predecessor though.
-Final Fantasy VIII - in my opinion this is a much better game than FFVII, however, it is still one of the weaker entries in the series. I really liked the quite complicated gameplay, and I also especially liked the dual-storyline going on with Laguna.
-Klonoa - this is a decent plattformer, although a bit easy. It has really good level-design, nice puzzles and interesting secrets.
-Tomb Raider - I think this is still the best game of its kind (of the ones I have played). The controls are a pain in the ass if you have played Mario 64 or anything that came after it, but I really like how the focus in the game is so much more on puzzle-solving and exploration than on gunplay, in stark contrast to later Tomb Raider-games (and clones, like Uncharted).
-Symphony of the Night - This game is of course awesome. However, I am a bit burned out on the fact that you are always traversing the - more or less - same castle in these games. Its great if you haven't played all the handheld ones, but if you have, its pretty similar to them unfortunately, and might be a bit boring. Its also really easy unfortunately.
-Rayman - This is a colorful and challenging plattformer that is probably a bit overrated since it was one of very few of its kind on the PS1, but its still a quite good game if you don't mind the slow tempo and the at times ridiculous difficulty.

So that's more or less the games I have played extensively on the PS1 outside of games like Tekken and Gran Turismo which in my opinion does not really hold up that well.