I don't know if it counts as a 'song' as such, but one piece of music which has always touched me since I first heard it as a kid is Beethoven's Moonlight Sonata. I spent hours as a kid on my cheap Yamaha keyboard picking out the notes from the music and can still play the first few bars I played it that much!
It astonishes me that something written by someone so long ago could write music which sounds so beautiful to this day. It also shows that there's something in humans that hasn't changed in hundreds of years! I can also imagine hearing 'Ode To Joy' in a Cathedral with the full choir being absolutely incredible.
It we're talking modern music, for me there's a few songs: Good Vibrations by the Beach Boys is almost perfect and I never, ever tire of hearing it. I won't mention The Beatles as there's literally dozens.