Phew...let's see what I can recall:
Atari: Grand Prix, followed by Pitfall.
NES: Super Mario Bros./Duck Hunt, followed by Marble Madness and Legend of Zelda II -- first game I ever beat on the system was Castlevania II
Sega Genesis: Krusty's Fun House, followed by Sonic 1. Only system I ever finished a Mortal Kombat on.
Apple IIe/PC: Mines of Titan, followed by Win, Lose or Draw -- first game finished was Incubation
Game Boy: Dr. Mario, followed by Super Mario Land and Tetris. First completed game was Kirby's Dream Land.
SNES: Super Mario World, also the first game finished, but was quickly followed by NBA Live '95, Link to the Past and Street Fighter II Turbo
N64: Goldeneye 007, followed by Killer Instinct -- first finished was WCW/NWO Revenge's Championship Mode
Playstation: Hardball 5, didn't actually finish a game on PS1 as I mostly played sports games
Gamecube: Resident Evil 4, also the first game finished -- obsessed over Smash Bros. Melee, though
Playstation 2: Need for Speed Underground 2, but finished Tony Hawk's Pro Skater 2 first. Quick shout-out to the Slim PS2, though.
Nintendo DS: Nintendogs!!!!
PSP: Lumines, followed by MLB and Metal Gear Acid
Playstation 3: Rock Band, which still haunts me to this day. First game finished on the PS3 was Journey.
XBox: Sigh...Dead or Alive Xtreme Beach Volleyball, though I don't believe I ever rolled credits on an XBox title. By this point, we have reached the time in my life where it would be almost impossible to complete a game.
XBox 360: Minecraft, followed by all the GTAV. Nothing 1000'd, though.
Playstation 4: Flower, followed by Sound Shapes. First game platinum'd was the 1.0 release of Rocket League.
Nintendo Switch: The Legend of Zelda: Breath of the Wild, followed by Splatoon 2. Super Mario Odyssey is the first finished Switch game for me.