The following are three pieces of music which evoke strong emotional responses inside of me, mostly good ones. All are from Playstation 1 games; 2 being from the same game.
They have always stood out as exquisite pieces to me, simply for the atmosphere and the climactic moment they bring.

R-Type Delta Soundtrack - 11 Awakening (Stage 6)
The piece starts with something dark, eerie, evil. This is fitting because the player spawns in a parasite/tumor overgrown world, perhaps the pilot's own world even.
2:52 is the part which evokes such a strong response in me (the sequence starts 2:34 and ends 3:30). It sounds very hopeful, but also fragile. Hopeful is a necessity when you're trying to save your world from an alien invasion, and the fragility might stem from the fact the game is a one hit KO type situation.
Then starts an awesome sequence which is pre-boss battle music where you fight infested starships, which is a miniboss I guess). There might still pilots be in there you once even knew.
The entire piece is basically 3 parts, I definitely prefer 2>1>3, but the entire piece fits so well.

This brings me exactly to the second memorable piece of music from the exact same game and it's from the next - and last - level.
R-Type Delta Soundtrack - 12 Fate (Stage 7)
A mostly symbolic, cathartic level. The music kind of repeats itself, but adds to the way the level is experienced: a beautiful wasteland is shown of what humanity - and more fundamentally, biology itself - once was.

The last piece of music is taken from Star Ocean 2: The Second Story. A realtime RPG with great combat systems, bosses, lore, crafting, items, etc.
A Quirk of Fate - Star Ocean: The Second Story OST
I always thought it was quite the complex game for a PS1, but I don't know how complex other RPGs were since I never really played any other.
This particlar rendition of the piece (because the same theme shows up several times up until now) plays when disc 2 is just entered. You have been transported to a new, alien world no one seems to know. But then one of your main character's party recalls she knows this place. She can feel it, but has no memory of it. The emotion the music conveys with the specific line the character says at that moment is absolutely one of the most evoking things - hopeful and tragic at the same time - I've learned to experience. I keep experiencing this exact feeling every time I listen to the piece. It's timeless.
Also, a lot of the boss battle music is exceptional (I think one was taken from a Final Fantasy game, but I can't recall which one). Check out this music for the fight with Indalecio, for example.

3 bonus pieces because they always bring me in such a great mood and I've used them several times as alarms to wake up to (my girlfriend hates them lol):
NES Batman, first stage
Mario Must Die (a romhack) - F-F-Fire!
Diagonal Mario - Track 1932 (a troll romhack)

I just had a strong urge to share these pieces today. I don't know if you know the pieces or not, but I do hope you will enjoy listening to them (again).