As others before, I can't really say which memory would be the one, but here's one of those I'm really fond of:
The year is 1992 and I have just reached the final level of Turrican II on my Commodore 64. At the beginning of the level, I find a hidden jetpack and start flying up. It's not an easy level; there's a plenty of monsters and gun turrets, but the game has trained me well enough in the previous worlds and levels. Plus there are many power-ups scattered throughout the level.
I fight my way through another tough wave of monsters, and - there are no further ones in sight.
A second passes. Two. Three. And then, at the bottom of the screen, something starts shifting up. A big arc slowly moves onto the screen. It's rougly a top of a big ring shape. As more and more of it gets visible, I finally realize what it is. A helmet. Giant helmet. Helmet about the size of my character.
My jaw drops. My lips start moving. "You... can't... really..." I hear myself whisper.
Without any sign of hurry, the behemoth slowly ascends into view. I can see the huge shoulders, chest with a built-in gun whose caliber alone is about half my height, jetpack the size of a two-story house. This is something larger than anything I've ever seen, a monster larger than I could ever even imagine. How could I even think of taking of something this big?
For the first time in my life (and still last to date), I feel pure, bone-freezing awe induced by a computer game.
Slowly, majestically, seemingly uninterested in my presence, The Machine passes by. It doesn't fit onto the screen - it's about 1.5 times taller.
The Machine has finally passed by, and I know we'll meet again. Probably very soon. My character knows this will be a true fight for the future of the universe, and winning it would equal a small miracle. My gaming self is still in a state of utter disbelief. The C=64 isn't supposed to be able to do something like this. No computer can! Ever!!!
About ten seconds later, I flew out of the huge cylinder and into the open. There, The Machine was waiting for me, and an absolutely epic battle ensued. But that battle was so hectic that I wouldn't have appreciated my opponent enough. The magic had happened before, when Manfred Trenz (the game's designer) gave me all the time to see, admire, and incredulously stare at The Machine. I later found out this bit was missing from the 16-bit versions, and even if the C=64 version didn't have the totally polished playability it has, I would still find it superior just for these few seconds of my nemesis heading for the final confrontation.
Well, I'm in my forties now, and gaming isn't the most important thing in my life anymore. I haven't played Turrican II for ages. But even now, almost thirty years later, I still got goosebumps when writing about it.