The first one was probably rolling the score on Joust when I was about 13. Once I cleared 10 million, I just dumped lives to end it. My Pac-Man killscreen made me feel accomplished, but it's meh. Anyone can do it. Finishing Pitfall! with a perfect score, both lives, and over 1 minute remaining is one of my favorites.
The number one accomplishment that made me really feel like a decent gamer, so far, is probably Yolympics 2017, when I finished 22nd, ahead of such luminaries as Riley, Fly, Exed, and Duaber, and in which I had 5 Top 15 scores and 1cc'd Twin Cobra II. Never grinded so hard in my life.
On a personal level, my best accomplishment was finally kicking the asses of 3 adults, playing Hyper Sports when I was 14-15. Every Saturday for about 6 months, a group of 4 of us would meet up at the mall at Aladdin's Castle, and even though I was the only one who had to have their mom drop them off, we'd play for hours on end. When I finally mastered my seizure-style button mashing technique on the swimming, I was finally able to bury them all consistently, after getting MY ass kicked for the first month and being laughed at and made fun of.