My favorite is beating Final Fantasy (NES) for the first time. Using my favorite config (Warrior/Red/White/Black mages) and going in to fight Chaos WAY early, I think I was only around lvl. 19 or so when the guide recommended 25ish or something.
I knew it'd be a slaughter. Chaos got my party down to just the warrior hovering at 2HP, no items left. "Well that's it" I thought as I chose "Fight" and... BOOM - Down goes Chaos! I ran out my house and did a lap around the block for that one.
Other favorites - my first AAA in Dance Dance Revolution... getting the 108 stars ending in Suikoden I... and another favorite although a recent accomplishment - No-Hit full victory in Mike Tyson's Punch Out!!
... yep... didn't get hit once. I've tried to redo it in front of an audience for witnesses but always get nervous and get nailed by that wonky Soda Popinski somewhere down the line. Grarg.