PS1: Final Fantasy VII, Runner up: Final Fantasy IX
PS2: God of War, Runner up: God of War 2
PS3: The Last of Us; Runner up: Ni No Kuni
PS4: Uncharted 4 (....for now) Runner up: Final Fantasy XV

PSP: Crisis Core: FFVII Runner up: Metal Gear Solid Peacewalker
PSV: Persona 4 Golden, Runner up:Gravity Rush

PS1: Crash bandicoot 3

PS2: God of War 2

PS3: Red Dead Redemption

PS4: The Witcher 3

PS1: Crash Team Racing
PS2: Grand Theft Auto: San Andreas
PS3: Dark Souls

PSP: Monster Hunter: Freedom Unite
PS Vita: Uncharted: Golden Abyss

Almost impossible to narrow it down to just one per gen, but any one of these are top candidates.

PS1 - Final Fantasy Tactics

PS2 - GTA 3

PS3 - Just Cause 2

and I know you said game of the generation, but I'm not leaving the Vita out in the cold

PS Vita -Uncharted: Golden Abyss

So fa, my PS4 pick might be EDF 4.1

Ps1: Final Fantasy VII
Ps2: Shadow of The colossus
Ps3: Heavy Rain
PS4: so far Bloodborne but it could be Horizon soon for me.

PS1: Gran Turismo 2
PS2: Grand Theft Auto: San Andreas and Ratchet & Clank: Up Your Arsenal (tie)
PS3: The Last of Us and Uncharted 2: Among Thieves (tie)
PS4: Uncharted 4: A Thief's End

PS1: Gran Turismo 2 (MGS and FFVII tied for second place).
Honorable mentions: Lunar: Silver Story Complete and Lunar 2: Eternal Blue Complete
PS2: God of War 2 (FFXII in second place)
PS3: Ni no kuni: Wrath of the White Witch (GoW 3 in second place)
PS4: I've had it for less than 4 months so I cannot rightly answer this yet.

I really don't understand why that game made such an impact on you...

PS1: Xenogears
PS2: Xenosaga Episode 3
PS3: Ni No Kuni
PS4: Dragon Quest Builders I guess... Easily the game I dumped the most hours into.

Fun thread idea!

PS1 GOTG: Tomb Raider (1996)
Tomb Raider features a smart and balanced blend of real-time combat and acrobatics, exploration, and puzzle-solving. The environments and challenges are varied, and the sense of adventure and discovery is invigorating.

PS2 GOTG: Ico (2001)
Ico introduces to gamers a huge, mysterious castle to explore and navigate, and lots of clever puzzles to solve. All of it is wrapped up in unique symbiotic gameplay between the hero Ico and his charge Yorda.

PSP GOTG: Killzone: Liberation (2006)
Before Guerrilla lost its way, it delivered an enjoyable and challenging tactical shooter exclusive to PSP. It mixes skill and strategy well, and offers a long list of mission types and boss encounters.

PS3 GOTG: Demon's Souls (2009)
The crown jewel of the PS3 library, Demon's Souls is a once-in-a-generation title. Punishing and cryptic, the game can and will turn off many players. Those willing to scale the steep learning curve will find an addictive title with stellar level and boss design, and a unique asymmetrical online multiplayer element.

PSV GOTG: Wipeout 2048 (2012)
One of gaming's great cruelties is that Sony Liverpool was shuttered shortly after releasing arguably it's greatest game. Wipeout 2048 is a title with huge replay value and stellar track design, and an inventive interpretation of online multiplayer.

PS4 GOTG: The Last Guardian (2016)
Fumito Ueda makes his second appearance on this list with The Last Guardian, a spiritual successor to Ico and Shadow of the Colossus. It's a clever mix of vertigo-inducing platforming, problem-solving, and an evolved version of the symbiotic gameplay introduced 15 years earlier in Ico.