I remember playing Ridge Racer and Battle Arena Toshinden back on a friends PS1 in 95 or something like that, but there were three games that really stole my heart, ironically, given their current status, Konami games:
Castlevania: Symphony of the Night. As a long time Castlevania fan from the NES days, SOTN blew me away with its detailed graphics, goat soundtrack, and Metroid style game design. I hadn't actually played the Metroid games at the time, so that style of gameplay was completely new to me.
Metal Gear Solid. This was the first game I played that not only made me feel like I was playing a movie, but made me feel like the creator was treating me like an adult. The complex plot, real world themes of nuclear responsibility and genetics, wrapped in a stylish, anime infused package had me drooling.
Silent Hill. I loved the Resident Evil series, and they were scary, but Silent Hill was the first survival horror game I played that actually, legitimately terrified me. The art design, the amazing soundtrack, the atmospheric design of the town and the nightmare fuel creatures that inhabit it catapulted the series as my favorite survival horror franchise of all time. Not to mention its cerebral, Jacob's Ladder-esque narrative had me puzzling over the mysteries of the town of Silent Hill for decades.
I'd also throw Vandal Hearts in there. It was my first SRPG, and was kind of a baby's first SRPG for me, but I remember me and my ex-brother-in-law staying up until 5 in the morning taking turns playing missions.
Square-Enix gets a lot of deserved praise for bringing some top tier games to the PlayStation 1, but I think Konami also deserves a lot of praise as well. The PS1 generation really seemed like a lot of Japanese developers really came into their own, with the PS2 generation having them perfect their craft.
I'm torn between whether the SNES or PS1 are my favorite consoles of all time. If probably give it to the PS1, mainly because of the three games listed above, and Squaresofts RPG output.