Not exactly a 'greatest,' but certainly a 'most memorable' moment, from more than a decade ago when I was still in Freshman\Sophmore level college. There was a pool hall\video game filled rec room in the basement of the cafeteria, where a number of (poorly maintained) arcade games resided. The particular one I was playing was Capcom vs. Marvel, and at the time (I got better) I absolutely
stunk at it. Wasted way more money than I should have playing that game, and I couldn't beat any of the regular players (and few of the irregular players, either). It was an extremely popular game, almost always played in versus mode, so I had never seen the final boss level.
I was waiting there later than usual, one evening, for my girlfriend (later fiance, later long-distance-relationship-didn't-work-out-ex-fiance. Utterly irrelevant to the story, I know), when I noticed the game console was strangely unoccupied. I only had enough cash on hand for one game, but I decided to try my luck.
I had finally, for the first time ever, played against the computer all the way to where I was about to see the boss level -- in fact, was in the middle of fighting the boss -- when one of the regular competitors came by to play. I was ticked off, because I had never gotten that far before, and wanted my chance to beat the game (with no continues) to end well... but I didn't think I had much of a chance at success, because the guy who joined in was -- as I said -- one of the regulars. Not just any regular, either, but the best player on campus, who owned every record in the high scores on that game (and just about every other game in the arcade). Knocking this guy off the console was the usual daily past-time for just about every one of the players in that arcade, regulars and irregulars alike -- people would follow him around from console to console just to get a shot at him.
But, like I said, I was ticked off. And while he made no mistakes in his gameplay, I somehow was just that much more motivated than usual, and walloped him in record time. Only time I ever beat the guy (on any game in that arcade), and he was practically in shock when he lost.
Then again, that's pretty typical of me. I had similar incidents when playing other games, as well. Take Chess against my brother, for example, who at the time was an FIDE-ranked Candidate Master (peaked at 2246 points on the FIDE scale; had to look up his former title, because he hasn't played competitively in a long time and I forgot what the title was; essentially means he was one of the top 20,000 players in the world. Me? I never entered an official competition, but I was told my level on the FIDE scale would probably top off around 700, had I entered competition, which would put me below what the usual just-starting-out ten-year-old amateur could accomplish). I was ticked at him (as brothers often were), and decided to try my luck with what I knew as the Paris Defense (not, apparently, it's real name, but
close. The actual opening I used was Paris
Gambit). It's generally considered a suicidal opening in chess, but heck -- I almost never was able to beat him, anyway, so I might as well play something crazy, right? I've never beaten my brother worse in any chess game before or since. (hence my inclusion of that particular chess opening in Training for the Job).
Word of advice if you ever play any games against me: Keep me happy, and chances are you'll cream me; I'm just not very good at games when I'm in too good a mood. Make me angry, and expect a devastating loss.
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Halo 2/Legendary on single player campaign (a lot harder than the other 2) 
Gears of War 1 (harder than the second one to me)/Defeating General Raam on single player campaign (after many different strategies i finally found one that worked for me) 
