The only song I can think of that I might not have lived, had I never heard it is
Happy Birthday to You. 
But on a serious note, there are so many great songs, there aren't any I couldn't live without.
That said, there are some that evoke intense feelings:
Fire and Rain - James Taylor
Kiss and Say Goodbye - The Manhattans
Us and Them - Pink Floyd
American Pie - Don McLean
Drift Off to Dream - Travis Tritt
Crazy - Patsy Cline (Pretty sure I would've come up with this even if you hadn't)
Dreams - Allman Brothers Band
Blues at Sunrise - Albert King and Stevie Ray Vaughan