I can't really speak for nostalgia, since I spent most of my time with PS1 just watching my big brother play Final Fantasy -series when it was still current-gen, but for whatever reason I've started playing a decent amount of PS1 games recently.
Parasite Eve II - Played this for the first time just this year, and while I usually try to be tolerant of other people's opinions, what the fuck is wrong with this games' reception? It's like Resident Evil but everything's better; more varied and distinct weapons, enemies and equipment, more secrets, solid story, better gameplay including far superior auto-aim and radar that helps you outmaneuver enemies, extra game modes, EPIC bosses, freakin' MAGIC SPELLS and more. I don't even like Resident Evil that much and I think this game is amazing!
Spider-Man - Pretty much the most you could ask for from Spider-Man on PS1, and I love how you can stealth kill enemies by jumping on their backs and just pound them in their heads. Never gets old.
Final Fantasy I - I like this one for it's simplicity with fair amount of strategy, and the graphics are simple 2D at its finest. The way you have to choose your permanent party at the start is also really cool and adds tons of replay value.
Tenchu: Stealth Assassin - Just plain fun stealth killing, especially if you're amused by old, overdone blood effects. Also, the cut-scenes are pure unintentional comedy-gold.
Silent Hill - Creepy though not downright scary, and the combat and exploration are really good for it's time.
