OoT and Waveracer saved the N64 for me. I bought it for Mario 64, never completed it. It wasn't all that fun to play. I had enough of it by the slippery ice stage.

The only one I can think of is Final Fantasy X for PS2. It's the only time I've bought a system on the strength of a single game.

SNES - The Legend of Zelda: A Link to the Past
N64 - Super Mario 64
PS1 - Final Fantasy VII
PS2 - Final Fantasy X
GameCube - Wind Waker
Xbox 360 - Virtua Fighter 5 Online
PS3 - Valkyria Chronicles

3DS - A Link Between Worlds

Switch - Breath of the Wild

Xbox for KOTOR

I bought the PS2 for Kingdom Hearts. PS3 primarily to play Marvel vs Capcom 3 and Street Fighter 4 with my coworkers online even though I had the games on 360. PS4 for Kingdom Hearts 3 and Spider-man (would have bought it or an XBox One for KH3, but Spider-man swayed me toward PS4. Dreamcast for Marvel vs Capcom, Sonic Adventure, and Soul Calibur. PSP for Mega Man Powered Up.

For Nintendo systems, there's really no one game that makes me buy them, cause I like a lot of Nintendo games. But, the game I was most hyped for on GBA was Golden Sun, Melee for Gamecube, New Super Mario Bros for DS, Mario Galaxy and Brawl for Wii. For 3DS and Wii U there wasn't really any one game that stood out that was known about at launch.

Ys 1 a PSP
Shenmue II and Jet Set Radio an Xbox
Bayonetta 2 and Xenoblade X a Wii U
Shenmue a Dreamcast

None

Super Mario Bros made me buy an NES.

But even Mario 64 and Zelda OOT couldn’t get me to be an N64.

Metroid Prime made me buy a GameCube.

At this point I’m Nintendo for life and I can’t see ever not buying a hybrid or hole console from there day one.

Sony would need to revive the Jak and Daxter franchise and show a commitment to returning to the PS2 era of game development, which simply isn’t going to happen.

I’ll probably never buy a Microsoft system. They pulled the hobby in a direction that was diametrically opposed to what I was looking for from the hobby and that’s difficult to reconcile this far in.

Super Smash Bros. Melee for Gamecube
Ocarina of Time 3D for 3DS

Of course, those were far from the only games I purchased for those systems, but those are the only times I can remember getting a system just because one game excited me enough.

NES - Super Mario Bros, Duck Hunt and Gyromite - The last two came with the system I got, but I really wanted to have Mario as well.
SNES - A Link to the Past - I actually bought the SNES on day 1, but I knew the next Zelda was coming in the next 12 months, and that was the game I wanted most.
Genesis - Sonic the Hedgehog - I was curious about what games the Genesis had, and Sonic was the one I was most eager to try.

PS1 - Final Fantasy 7 - I saw the marketing for it and I was like "Woah!  They skipped 4, 5 and 6 and went all the way to 7!  This game looks awesome, and i have to play it!"
PS2 - Didn't buy.  It was a gift.  But if I did buy it, it would have been for Final Fantasy X.

GBA - Played a little Final Fantasy Tactics Advance on a friend's system, and I had to have it for myself.
DS - Final Fantasy Tactics A^2.

PS3 - Bought it in 2008, but I was anticipating the release of Final Fantasy 13.  Yeah, this is where Final Fantasy stopped being my favorite series.  The game becomes really fun after about 20-25 hours, but you shouldn't have to play that long just to get to the good part.

Wii - Got it in 2009 for New Super Mario Bros. Wii.  It's still my favorite Wii game.
Wii U - Nothing.  I didn't want to buy it, but my kid really wanted one, so I caved.

PS4 - Combination of Dark Souls 3 and Bloodborne.  Turns out that Bloodborne wasn't that great, but DS3 was really fun.

3DS - Bravely Default.  Game was decent, but I never finished it.
Switch - The Legend of Zelda: Breath of the Wild.  Day 1.  Last time I bought a system at launch was the SNES.  And while I regretted buying the SNES too early, I had no regrets about buying the Switch.