It's got to be Starwing for the SNES. I spent months on end trying to beat the final levels on Hard mode. And one day, it just clicked. I was on my way to Andross with less than half my energy bar and on my last life, certain that I couldn't beat him.
"I knew you might make it eventually..." Andross taunted me, "Your father was a stubborn fighter too. But this will be the McClouds' last battle!" I entered the same combat arena I'd been through on Easy and Medium mode. Andross's gigantic metal simian head formed in front of me, familiar to anyone who's seen the Andross assist trophy in Smash Bros. Brawl. I blasted the face apart twice, and shot everything I had at his core, twice. Then he reformed for the final stage...
and morphed into a bull. He shot plasma balls at me as well as everything else. I was in a pure shmup trance, dodging and shooting without a second thought. The bull's head broke apart, and there was the core again. I shot as hard as I could, and it exploded.
Suddenly, it clicked. I had beaten Starwing.
I had beaten Starwing!!
YES!!!!!!!
It was the first game I ever played all the way through; the first game I ever really wanted to. And it took a long time, but I beat it. I beat it but good.