I don't know where to start. I love a lot of acoustic songs. But those recorded acoustic songs (not acoustic covers or acoustic versions)... I'd say:
'Love me', Colin Raye. Just beautiful, and very sad too. It's a top song at funerals from what I've heard.
'When the Children cry', White Lion (the guitar solo is the only electric guitar part in the song, the song is still driven by flat-picked acoustic guitar)
'Hurt', Johnny Cash. My kind of song. Even though it is a cover, I like it very much. But then, I like NIN's original 'Hurt' just as much. Since Trent Reznor plays it on a piano, I guess that would also count as an acoustic song.