SMB3 - I GOT THAT ISH ON VIDEO, SON! Except I don't know if the video got damaged and I don't have a VCR to check.
The story behind it: At about age 10, my cousin and I had been trying relentlessly to beat the game earlier in the day, after many unsuccessful attempts, he had to go home and I stayed playing, so I turned on the VCR to record so I could show him how far I got when he wasn't there. I got all the way to the end and defeated Bowser on my second attempt. Neither of us had ever gotten that far in the game. It was a glorious moment indeed!
SMB2 - Couldn't believe that the whole thing had been a dream! I didn't actually beat SMB2 until after 3.
SNES Star Fox - Always wanted this game (and an SNES) when I was a kid but never got them.
Later, when the emulator scene was in effect and I had a decent computer to run them, I got me a "backup" of said game and with a controller attached to my PC, I proceeded to take on the challenge that had been denied to me previously in life. The ability to do save states certainly helped.
SM64 - What can I say? An awesome feeling of accomplishment after defeating Bowser in the 3rd dimension!
Half Life, PC - Couldn't figure out what that freaky white thing was at the end, but I defeated it. I had to play this game in secrecy as my parents didn't approve of violent games at the time so I only played it when no one was home or asleep. I remember one time my mom caught me playing Half Life Death Match against a friend and she immediately came up to me hugging me saying how I shouldn't be playing violent games so I don't turn into a violent person or something like that. Now my mom doesn't give a crap, my little brother's play things like CoD and Gears of war and stuff... And we all turned out ok.
Doom - Never got to play Doom when it first came out, but later on, a friend let me borrow one of those CD's that came with like 200 games on one CD or something ridiculous like that. Most of them sucked, but there was some on there like Doom that were ok. I again played this one in secrecy until I beat it. I don't know, there was something about beating a game that you knew your parents wouldn't approve of, kinda like looking at boobies when you knew you "shouldn't".
Star Fox 64 - Talk about button-masher! Just recently beat this one, actually. I think it's paved the way to arthritis in my wrists.
There's more, like God of War I and II. Now I gotta beat III, but I don't have a PS3. Bioshock - PC, the twist at the end was like WTF? And more.. but I'm typing this at work and I'm gonna have to go soon, so I'll have to cut it short.