Pokemon Red? I guess, or that really old Superman game on Sega Genesis, Superman Doomsaday or something. It was the one where you got to fight as different versions of Superman.
Ninja turtles on NES. Around age 5. That swimming level.. ugghhh. I haven't played that crap in like 20 years and I'm pissed off just thinking about it
The first game I wholly finished was Super Mario World SNES. But I beat the NES Marios, just not every world.
Played the shit out of Atari Frogger and Pong, but I don't think there was an end to either of those. I don't recall finishing Raiders of the Lost Ark either.
Beating Star Fox was probably the most "fun" I had during that era. Remember polygonal graphics? Those were revolutionary and amazing. :D