Super NES - Chrono Trigger
64 - Ocarina of Time
PS2 - Dragon Quest 8 (honorable mention: Kingdom Hearts 2 and 1)
PS3 - Ni no kuni
Wii - Xenoblade (unless it doesn't count because of the new 3ds version?), if not then Skyward Sword
PSP - Kingdom Hearts BBS
DS- The World Ends with You
3DS - Kid Icarus Uprising

Edit: PC - Age of Empires 2

I've never had a Sega console or a Xbox.

Snes - Castlevania 4

N64- OOT

Gamecube - Smash Bros Melee

Ps2 - Shadow of colossus

Xbox 360 - Halo 3

PC - Guild Wars 2

3DS - Monster Hunter 4 Ultimate

Wii - Mario Galaxy 1

Wii U - Mario 3D World

Takes some thought this thread but here goes, I'm pretty sure pretty much nothing I played on my Atari was an exclusive, so I'll start with the NES in chronological order:

NES: Super Mario Bros. 3
Mega Drive: Streets of Rage 2 (if releases in the Arcade are also discounted; Castle of Illusion)
SNES: Super Mario World 2: Yoshi's Island
PS1: Kula World (US: Roll Away)
N64: The Legend of Zelda: Ocarina of Time
Dreamcast: Soul Calibur (if no Arcades (I'll need to skip a whole bunch of games); Shenmue)
GameCube: Luigi's Mansion
Wii: Super Mario Galaxy
PS3: Uncharted 2
WiiU: Mario Kart 8
PS4: Uncharted 4

I played the original XBox, but no exclusives.

GB/C: The Legend of Zelda: Link's Awakening
GBA: Metroid Fusion
DS: Professor Layton & Pandora's Box
3DS: Resident Evil: Revelations

If PC and Mac count for one, then SimCity 4, otherwise:

PC: Rollercoaster Tycoon 2
Mac: Step-On-It (probably the most niche game ever, but everything else I played on Mac was also on consoles or PC/DOS, still a very fun puzzle game)

PC - Warcraft 3: The Frozen Throne

NES - Gun.Smoke
SNES - Secret of Mana
N64 - Hate that console
GC - Windwaker
Wii - Xenoblade Chronicles
Wii U - Xenoblade Chronicles X

PS3 - Last of Us (at first I was kind of dissapointed, but after I have completed it.. it's grown on me tenfold, also I beat it on PS4 but since it's a PS3 game.. )
PS4 - Only beaten multiplats and remasters.. so nothing to write in here.

PC: I want to say Mount & Blade: Warband but that's on PS4 now. And Age of Empires 2 also doesn't count because of it was released on PS2, right? So I'll something lame like Pillars of Eternity then. No wait, make it the original Age of Empires.
Xbox: Halo
Xbox 360: Halo
Xbox One: Halo
XB Scorpion: Halo
PS1: God of War
PS2: God of War
PS3: Demon Souls
PS4: God of War
NES: Zelda
SNES: Zelda
N64: Zelda
Gamecube: Zelda
Wii: Zelda
Wii U: Zelda
NX: Zelda

PC: Half Life 2
Master System: Alex the Kidd
NES: Super Mario Bros
Mega Drive: Streets of Rage 3
SNES: Zelda A Link to the Past
Mega CD: Ground Zero Texas
N64: Zelda Ocarina of Time
Saturn: Panzer Dragoon 2
PS1: MGS
Dreamcast: Shenmue 1+2
Gamecube: Rogue Squadron
PS2: Devil May Cry
Xbox: Otogi2
PS3: GOW3
Xbox360: GeoW 2
WiiU: Super Mario World 3D
PS4: Bloodborne

Gameboy: Tetris
Game Gear: Columns
Atari Lynx: Rygar
PSP: GOW GoS
Vita: Gravity Rush

PC: Civilization (Franchise)
SNES: Donkey Kong Count
N64: Zelda Ocarina of Time
PS1: Final Fantasy IX
Gamecube: Smash Bros Melee (Tales of Symphonia if we go NA only)
PS2: Suikoden 3
PS3: Last of Us
Xbox360: Ace Combat 6 (Tales of Vesperia if we go NA only)
Wii: Donkey Kong Country Return
WiiU: Mario Kart 8
PS4: Pending
Gameboy: Pokemon Yellow
Gameboy Advance: Megaman Battle Network
DS: The World Ends with You or Dual Strike
3DS; Pokemon X and Y
PSP: N/A (Never find a title that was exclusive that would be a favorite per say)
Vita: N/A (Never find a title that was exclusive that would be a favorite per say)

GB: Pokemon Crystal
GBA: Pokemon Emerald
DS: Pokemon Mystery Dungeon: Explorers of Sky
3DS: Fire Emblem Awakening

NES: Super Mario Bros.
SNES: Chrono Trigger
N64: Super Mario 64
GC: Paper Mario: The Thousand Year Door
Wii: Fire Emblem: Radiant Dawn
Wii U: Super Mario 3D World

PSV: Gravity Rush

PS1: Final Fantasy VII
PS2: Persona 4
PS3: Infamous

PC: Civilization V

NES - Mega Man 2
SNES - Super Ghouls`n Ghosts
N64 - Donkey Kong 64
GC - Eternal Darkness
Wii - Punch-Out!!
WiiU - ZombiU
GB - Mega Man V
GBC - Wario Land 3
GBA - Mega Man Battle Network
DS - Rhythm Paradise
3DS - Ultimate NES Remix

PS1 - Crash Bandicoot
PS2 - Maximo VS Army Of Zin
PS3 - Demon`s Souls
PS4 - Bloodborne
PSP - Ultimate Ghosts 'n Goblins

Xbox 360 - Dead Rising

SMS - Alex Kidd In Miracle World
SMD - Castlevania: The New Generation

PS1: Crash Bandicoot 3
PS2: FFX
PS3: Valkyria Chronicles
PSP: Dissidia FF
PC: AoE 2

Honorable mentions:

PS1: FF VIII
PS2: Persona 3 FES
PS3: Demon's Souls
PSP: Persona 3 Portable
PC: Diablo 2, Starcraft1/2, CIV 3