I posted elsewhere about Portal… It has a great song from the ending… spoiler don’t watch the video if you want to experience it on your own (after all these years I doubt there are too many oldsters this applies to… but maybe some younglings haven’t played it yet.)
Are there other great songs I’ve probably never seen because I haven’t completed enough games 
Certainly Super Mario Bros.
That 8-bit vibe, huh? I feel that. That reminds me that there were a lot of great MOD/tracker demo songs in my Amiga days.
For me, it is the music for Frontier: Elite II on the Amiga
This is tough, because I love music, and I love video games (especially ones with good stories) but if I have to choose, it would probably be Final Fantasy VIII - Eyes On Me. It’s an enjoyable song even if you haven’t played the game, but it’s particularly moving and stirring if you have, and there are little hints or leitmotifs (e.g. Waltz for the Moon) leading up to it at the end. Pops into my head from time to time.
One of my favorite games ever, just after Chrono Trigger, and possibly tied with FFVI (maybe a hair ahead of FFVI).
I would either have to go Final Fantasy or Zelda. Soundtracks for all their games are usually good.
Zelda SNES 
What a great game that was!
Zelda’s lullaby. Ocarina of time
Tetris. Always plays in my head when I’m packing the car for a family trip.
More recently, Celeste used it’s soundtrack amazingly well to convey the story and the emotions of the character.
Fallout 2. The intro to that game introduced me to Louis Armstrong
I noticed you said memorable, instead of favourite or best of all time. That’s a far less loaded topic to get into.
There are a ton of great games with great soundtracks that it’s really impossible to hear them all. There were a few good ones already mentioned in this thread, so I won’t repeat those. I’ll just share some stuff I particularly like.
The Legend of Heroes: Trails in the Sky Second Chapter - The Whereabouts of Hope
Spellbound (Barry Leitch Remix)
I hope some of you enjoy them too.
Any of the Star Wars games…
LOL you just reminded me of the time I asked my Google Home to play “The Imperial March” expecting something from the movie soundtrack, and instead got this:
FF for sure. Nobuo Uematsu is a fantastic composer.
Also check this out:
Most memorable? Possibly the Oblivion theme - seeing the city in the opening cinematic and hearing the music was pretty amazing.
Others favourites: I listen to a lot of game soundtracks while working so enjoy the calmer ones.
The NES Mega Man Games had some great music. I actually still use some of those tracks for my ringtones on my phone today. I also have one of the Zelda tracks as my alarm to wake up in the morning.
Portal “Still Alive” is awesome. I actually have that song on my playlist.
While not a “song” but an instrumental, the original Elite 1984 music sometimes gets stuck in my head. It was a pretty amazing score for an 8-bit game on Commodore SID chip.
More recently I find certain tracks from Nier: Automata to be pretty amazing.
Oh, and let’s not forget the opening track to Skyrim! Playing that on the big home theater speakers is epic!
I was tempted to post another of Jeremy Soule’s works, Total Annihilation, but that’s a case where the entire soundtrack is epic; not as “memorable” in the sense of being catchy or an earworm as some of the classic console games.
Of The Elder Scrolls games, the Morrowind intro still takes the cake for me, that tune always randomly enters my head:
And in other RTS intros, Hell March 2 (all of Frank Klepacki’s music is awesome, though):
Generals, one of the other games in the C&C universe also had memorable tracks, particularly on the GLA side (that game was fairly prescient, with GLA being very ISIS-like more than a decade prior to its rise):
In more recent games, Prey (2017) was an “instant classic” for me:
And another Mick Gordon soundtrack, which obviously is paying an homage to the original DOOM.
The spiritual progenitor of Prey:
And in unreleased games, this track is in my “Your Ass Is Dragon” playlist (my workout playlist 
). Not sure if it counts since it might just be a trailer track:
Love the Morrowind soundtrack. Peaceful Waters is the standout for me.
Oddly, having a bit of alternative console and computer in the 80’s/90’s with amiga and Sega Saturn.
Like:
Battle Squadron (amiga ) https://youtu.be/37l56vc8PIY
Startust ( Amiga ) https://youtu.be/l9xOZ4jtg8w
wipeout (saturn/playstation) https://www.youtube.com/watch?v=5c2fm9TJClw
Tempest 2000 (saturn/atari) https://youtu.be/GMXvQZmm_Yg
closer to dance/techno music. After all theses years I gravitate around theses fast paced soundtrack.