Final Fantasy IV/II(US): Rydia defeating Octomammoth with a regular attack that did 1 HP of damage.
Final Fantasy VIII: Defeating the Omega Weapon on the first try, but not without trouble. Using a party of Squall, Rinoa and Zell, Omega used its Terra Meteor (been a long time, forgot the exact name) attack and wiped out the entire party, only to cue a revival by Phoenix. After another Terra Meteor, the party was wiped out again, but Angelo revives Rinoa. She didn't last long enough to revive the party and got KO'd again, only for the whole party to be revived by Phoenix a second time. The third time Omega cast Terra Meteor, it DID! NOT! HIT! Squall a single time. I managed to finish Omega off soon after that.
Kickle Cubicle: The first attempts at completing the final level in the game ended up taking about 45 minutes of trying, and still several attempts in later practice runs. When doing the final recording for the Let's Play of the game, I cleared it on the first try.
Mario Kart 64: Knocking Peach off the course on Rainbow Road and, for some reason, she did not get brought back onto the course. Her time at the end of the race was the full maximum on the clock.
New Super Mario Bros. Wii: Beating both That One Level and Scrappy Level 8-1 with all of the Star Coins without dying on the first try just by charging through the level almost without stopping.
As such, also beating 8-4 on the first try with all Star Coins, but with a much more conservative approach to the level and nearly running out of time.
Real Life: Of course nothing earth-shattering but we all have moments we're insanely proud of.
First place in the Annual High School Mathematics Exam (ASHME) two years in a row; both times by 1 point.
Receiving a standing ovation in the middle of a live performance of a Princess Bride spoof for doing a dead-on impression of the Impressive Clergyman.
Scott Pilgrim vs. the World: Defeating the Twins in Level 5 at the same time by getting thrown against the wall and ricocheting back to slam into them and deal the final blow!
Super Mario Kart: Winning 1st place in the final race of a 150cc Mushroom Cup run with a .001-second lead. I've also learned that getting the exact same time will still give the human player the win, but that was a different situation.
Tales of Symphonia: Due to a poor choice in which route to take when going for Lloyd's Eternal Apprentice and the I Hate Gels! titles, my team was vastly unprepared to fight Sheena in the Tower of Mana and Kvar in the Luin Ranch even before going to Palmacosta. As such, they were wiped out quite easily except for Lloyd, whom I was controlling. I managed to defeat both of those bosses single-handedly using Lloyd's initial weapon while being 10 levels below the recommendation!
Touhou: Concealed the Conclusion: Even though it was on Normal, doing a run that resulted in not taking a single hit, capturing every spellcard, and grazing so much as to have over an hour of time to fight the final boss is something I've yet to top.
Touhou: Imperishable Night: It being the very first Touhou game I played, I managed to get the Normal Ending for the Scarlet Team the very first time I played (did have to continue once though). The entire feeling of overcoming it like that was so overwhelming that I was laughing hysterically for well over half an hour after finishing the game. Thankfully almost no one was in the dorm (this was at college) at the time.
Wild ARMs 3: The first time I played through the game, I found the Secret Garden and entered the Decaying Labyrinth about 10 levels too early. As such I was very nearly wiped out by the boss, but at the end only Jet survived to finish the battle. He was down to 1 HP (due to the cursed ring that the party needs to destroy) and I still had to escape the dungeon. In every battle, none of the attacks hit him and I managed to clear the dungeon.