PS1: Tomb Raider series, Twisted Metal 2 (specifically the NY level)

PS2: Ico, GTA3, SOCOM series

PS3: Destiny, The Last of Us, Uncharted series, Heavy Rain

PS4: Destiny, Days Gone, Uncharted 4

PS5: Don't have - there's no game out yet that compels me to buy the system

To be honest, I didn't care much for PS1, so, off the top of my head, one of these that I know were on PS1 as well, but I actually played them on PC back in those days:

Tomb Raider 1
Legacy of Kain: Soul Reaver
Oddworld: Abe's Oddysee

PS1: FF7
PS2: GoW 2
PS3: Uncharted 2 or TLOU
PS4: God of War or Bloodborne
PS5: Astro Bot

For Playstation 1:

- Suikoden 1 & 2
- Final Fantasy 7, 8, 9, Tactics
- Resident Evil 1, 2, 3
- Castlevania: Symphony of the Night
- Metal Gear Solid
- Chrono Cross
- Tekken 3
- Megaman X4
- King of Fighters 97

Metal Gear Solid.

I havent played enough games on the PS1 - majority of the games from PS1 I played were from remasters or re-releases such as Metal Gear Solid and Street Fighter Alpha 3.

I would say Final Fantasy IX is my favorite from PS1.

As for the rest....

PS2: Kingdom Hearts II

PS3: Final Fantasy XIII-2

PS4: Final Fantasy XV

PS5: Astro's Playroom (the only PS5 game I've played lol)

PSP: Kingdom Hearts Birth By Sleep

Going to separate it out by franchise for the poll. Looks like the games that have 5 votes or more so far are 

• Crash Bandicoot
• Final Fantasy
• Metal Gear Solid
• Resident Evil 
• Spyro the Dragon 

These ones have 4 mentions

• Tekken
• Tomb Raider
• Tony Hawk

These have 3 mentions

• Twisted Metal
• Castlevania
• Legacy of Cain: Soul Reaver
• Suikoden
• Chrono Cross

I'll adjust the poll accordingly to accommodate whichever franchises get the most votes. To streamline, I did only count the franchise instead of multiple votes for the same franchise. 

A lot of great games on PS1, but if I had to pick the one that had the biggest impact in my gaming life, I would choose Crash Bandicoot 3: Warped.

FF IX, never owned one myself. Fear Effect 2: Retro Helix was good too.

When PlayStation launched, I wasn't born at that moment. But it's still honest answer.