There's no single answer, but this will always have to be in the discussion.
If you're into jazz at all, I'd check out some versions of "I Fall In Love Too Easily" (here is one by Keith Jarrett) and "My Foolish Heart" (Bill Evans' specifically).
In the classical world, I personally love Chopin's Nocturne in E minor, his Nocturne in C-sharp minor, and his Prelude in E minor. Debussy's Arabesque No. 1 is also one of my favorites. I'd also add Mendelssohn's Songs Without Words, specifically this one.
I realize that this is way more than one song.
Moonlight Sonata for me. Heard it for the first time when I was in elementary school and spent years trying to find out the name/composer.
But they're all beautiful so why not.
My aunt who is a retired opera singer always tells me about how my grandmother used to play it as her encore during her concert tours in Europe. I always get the same question when the subject of my piano playing comes up: "Do you know La Campanella already?". Apparently I am nothing until I can play that flawlessly.