I have to say really no. I usually wait until there are five or six titles I want for the system before buying it. The only "exception" are Nintendo's consoles since I know that Nintendo will supply those enough titles to make the console a worthwhile investment.

Anything made by Fumitu Ueda
or an Atelier game

Legend of Zelda...pretty much any of them. Gonna get tue NX due to BotW alone.

Everything exclusive from Naughty Dog.
Souls/Borne and GTA series probably too.

PSOne  = Final Fantasy VIII and Tales of Destiny

Xbox = Halo CE, Dead or Alive 3, Phantom Dust and Shenmue II

Xbox 360 = Halo 3, Tales of Vesperia, Final Fantasy XIII, Lost Odyssey, Dead or Alive Xtreme 2 and Oneechanbara*

Xbox One = Final Fantasy XV, Ryse, Psycho Pass, Halo 5 Guardians and Scalebound

Nintendo GameCube = Star Fox Assault, Metroid Prime, 1080 Avalanche

You'll notice something particularly odd regarding my choices. 

*The original because D3 sucks... 

And ToV 2 or any JRPG for a Microsoft console since they usually have more polishing than even Shitzzard and Nintenshit games... 

I never buy a console if it only has 1 game I'm interested in but I can list the games which have been the deciding factors in my hardware purchases.

Game Boy: Wario Land: Super Mario Land 3
PlayStation 1: Gran Turismo/Tekken 3/Crash Bandicoot
Game Boy Color: Pokémon Yellow
Nintendo 64: Super Mario 64
Gaming PC: Command & Conquer: Tiberian Sun/Half-Life/Diablo 2
PlayStation 2: Gran Turismo 3
Game Boy Advance: Pokémon Ruby
Xbox 360: Dead Rising
PlayStation 3: Uncharted/Valkyria Chronicles
Wii: Super Mario Galaxy
PlayStation 4: Driveclub/The Last of Us Remastered
PlayStation Vita: Minecraft/Wipeout 2048
Nintendo 3DS: Pokémon SoulSilver
Wii U: Super Mario 3D World/Super Smash Bros. 4
Oculus Rift: Half-Life 2 VR

If Dead Rising 3+4, PvZ: Garden Warfare, Rise of the Tomb Raider and Quantum Break had stayed exclusive I'd have an XB1 as well.

Edit: 12/23 are Japanese games. More than I thought actually

It's hard to say for me, because that one game could be anything. Heck, I bought a PS2 just for DDR. Whatever game that looks really fun to me is something that is worth buying a console for.

Nope

Prior to this generation I was contemplating not buying into consoles any more, going exclusively PC. But then I played The Last of Us, that game pretty much singled handedly convinced me to buy a PS4. As long as Naughty Dog are making games I'll buy the system they release them on.

I bought a PS4 for Persona 5 and Final Fantasy XV (PS3 YLOD). Even if those games didn't exist I would get a PS4 to play FFVII earlier. I can play most multiplats on PC. So Final Fantasy is definitely that series for me.