It doesn't have to be the first PS1 game you played or saw but the game you could not take your eyes away from. You had to play. Played/saw that made you need this console.

Final Fantasy 8.
I enjoyed the story... And the visuals to me felt more impactful over Final Fantasy 7's character design or the regression I personally perceived in Final Fantasy 9 character design.

The seamless transition from gameplay to FMV was just icing on the cake back then and was seriously impressive.

To be fair though, I never played many games outside of FF8 on the PS1, I was strictly PC at the time... Something to be said about having a Voodoo 2 instead of the PS1 blocky visuals or the Nintendo 64's blurry Vaseline output.

Though the JRPG's on the console did interest me.

FF7 for me. It was the first time I realized gaming could tell a better story than Hollywood. The size, scope, characters and story were just something else.

FFT makes my list as we. The level of strategy was simply superb. I want a remake so bad.

Played Crash on PS1 as one of the first 3D titles and that was impressive. Felt like I played out one of the Saturday morning WB-cartoons.
Never had much access to a PS1 but Tekken 3 and Tony Hawks pro skater 2 was played quite a bit when visiting friends.

Probably like Metal Gear Solid the most, but it did not have the same impactful first impression on me like Crash, Tekken or Tony Hawk.

I first played a PS1 at a Kiosk with an ESPN sports game. The game that blew me away was G Darius.  It's still a great game but this was my first Darius game. It was at a local game store in the mall across from the Arcade. Older games might remember a 13-inch TV with massive fold-out speakers. They had the game playing on that. I got to the first fish boss. Seeing 2D backgrounds mixed with at the time gorgeous 3D really gave a great impression on me even tho I played other PS1 games at a friend's house before.

Gran Turismo. It was mind blowing experience. It was my first PS1 game and I'd never seen anything like it

Not really sure what game I first got into with the PS1.

But the game that gave the biggest first impression was definitely FFVIII. While I was a big fan of FFIV(6) and FFVII, FFVIII was the first game I felt hyped for. When I started the game and the music starts with the amazing animations, I was hooked.

Resident Evil. I'd never seen anything like it - the atmosphere and the sense of discovery as the story slowly unfolded. It was an incredible experience back then.

None of the above, I guess.
I didn't care for PS1 much. Everything that was my cup of tea was on PC as well.

Games that piqued my interest were Grand Turismo, but PC had its fair share of good driving games (like Need or Speed and later Colin McRae Rally), and Tekken, but Battle Arena Toshinden was on PC as well, so those didn't make me go for PS1.

Croc: Legend of the Gobbos.
It was the first 5th gen game I ever saw in action; I was used to SNES games at the time, and the only 3D game I'd ever played before was Starfox.
The jump to fully textured levels that you could explore in any direction blew my 9-year-old mind.