So you at least commiserate with my distaste of its use in this context. 

In all fairness, to answer the question:

NES: Super Mario Bros

SNES: Super Mario World

PS1: Final Fantasy 7

PS2: Smugglers Run or ATV Off Road Fury (Obviously these wouldn't end up my favorite of the system...but they did get me hooked on the system when I played them every night at the video store I worked at)

Gamecube: Metroid Prime (This title very much brought me back into the Nintendo fold)

Wii: Wii Sports, Twilight Princess

WiiU: I'd have to say this generation didn't have that one title announced that I would buy it for.  At this point I just know they're going to release updated versions of games I love.  Arguably, the thought of seeing Mario in all his HD glory was a major draw...but I'm really still waiting for Metroid.  Metroid is why I buy Nintendo consoles...and its exclusion from the N64 lineup was why I didn't bother to participate in that generation...though I did play the hell out of Ocarina of Time and Super Mario 64.

PS1: Crash, Spyro, MGS
PS2: R&C, J&D, MGS3, GTA3
PS3: R&C Future, Uncharted, LBP, MGS4
PS4: Infamous, Uncharted 4, Ratchet Reboot

N64: OoT, Mario Kart 64
Wii: TP, Mario Galaxy, Brawl
WiiU: Mario Kart, Smash

Wow guess I always kinda get my systems for the same devs/franchises for about 20 years lol. Now to clarify I do play LOTS of other games on these machines but these games were there when I started or announced and already had me excited.

Halo, Gears of War and The Last of us.

Atari 2600 -- Seaquest. It was just plain and simple fun.

NES -- Double Dragon II. First game I ever "mastered", where I could just about beat it with my eyes closed.

SNES -- Final Fantasy VI. This is the first game changer for me. I was burned out on the same ol' platformer and brawler type games that seemed to make up 90% of the titles back then. I didn't really care about gaming very much. Then, by pure chance, I came across FF6 in a used game store that I was only in because a friend wanted to pick something up. I didn't even know there were more of the Final Fantasy games beyond the first one, which I'd owned and loved on the NES. It completely grabbed hold of me in a way no game had ever done before.

PS1 -- Final Fantasy Tactics. I bought it because it was a Final Fantasy spin-off but who knew it would probably become my favorite game ever?

PS2 -- Gran Turismo 3: A-spec. Another game that just blew me away and owned my life for months.

Dreamcast -- SoulCalibur. This game ruled so hard at the time, it's hard to even explain. It was such a magical leap forward that I will always think of it as the best fighting game ever made relative to the era it was released in.

PS3 -- Valkyria Chronicles. First PS3 game I beat and still one of the best on the console. I loved the story, the characters, and the game-play.

PC -- World of Warcraft. I spent years playing that game. I still think it's the most monumental achievement in the industry. It's absolutely massive, with a breathtaking amount of content. Even people who do not like MMOs should respect the amount of work Blizzard put into WoW.

Genesis: Sonic series.

GB: Super Mario Land 1+2,Pokemon RGBY.

Gba: Pokemon RS.Mario kart,Sonic Advance.

DS: Advance Wars Dual Strike, Mario Kart, Pokémon DP.

3DS: Pokemon XY/ASOR, Fire emblem awakening, Smash Bros, Tekken 3D.

Wii: Wii sports, Zelda Twilight Princess, Super Mario Galaxy, GC games.

WIIU: Smash Bros, Mario Kart 8, Xenoblade Chronicles X, Pokken Tournament .

PS1: Tekken,Resident Evil, Crash Bandiccot,Spyro,Gran Turismo,Final Fantasy VII.

PS2: Tekken Tag Tournament, Grand Theft Auto III,Final Fantasy X.

PS3: Kingdom hearts HD collections, Tekken Hybrid/Dark Resurrection, Dragons Crown, Ratchet and Clank series.

PSP: Crisis Core: Final Fantasy VII,Kingdom: hearts birth by sleep, Monster Hunter Freedom Unite, Ratchet & Clank: Size Matters.

Xbox 360: Blue Dragon, Gears of War, Halo 3,Fable 2, Original Xbox games.

Xbox One: Ryze, Forza Horizon 2, Sunset O, MGS: Ground Zeroes, Dragon Age: Inquisition
Xbox 360: Halo 3, the Fable serie, Alan Wake, Mass Effect (all of them)
Xbox: Halo, Rainbow Six, Fable
PS3: Metal Gear Solid 4
NES: Mario Bross 3
SNES: Super Mario World, Mario Kart
Sega Megadrive (Genesis): All the sonic games
GameCube: Metal Gear Solid: Twin Snake, Mario Kart: Double Dash
Gameboy: Super Mario Land, Tetris
PSP: Jeanne d'arc, Final Fantasy Tactics
GBA: Final Fantasy Tactics Advance
3DS: Metal Gear Solid 3 (really awesome on this console)

PC: Diablo 2, The Settlers (the original, I played like crazy), Dune 2 (RTS)
Atari STE: Maupiti Island, Vroom, Dune
Amiga: Black Cauldron

Those take me back.  Double Dragon II was certainly an inferior experience to what we fell in love with at the arcades...but still very much a revolution in gameplay.

Regarding Seaquest...actually that game very much seemed like an App.  Shoot, shoot, shoot.  No real end...just high scores and faster enemies.

Super Nintendo - Super Mario World
Saturn - Virtua Fighter Kids
Dreamcast - Crazy Taxi
Playstation - Final Fantasy VIII
Xbox - Halo 2
Playstation 2 - Final Fantasy X
Xbox 360 - Lost Odyssey
Playstation 3 - Final Fantasy X|X-2 HD
Playstation Portable - Final Fantasy Tactics: War of the Lions
Wii-U: Super Mario 3D World
Playstation 4 - Destiny (this will change to Final Fantasy X|X-2 HD when it is released)
Xbox One - Halo: Master Chief Collection
Playstation Vita - Final Fantasy X|X-2 HD
PC - Final Fantasy VIII
Cell Phone - Final Fantasy III

X1 - AC Unity bundle + Halo for $349.