This was such a good idea for a thread. I am finding all sorts of songs I never really listened to before thanks to y'all.
I wanted to add 2 more:
This cover of Tom Waits' "Time" done by Tori Amos makes me cry. I love Waits, but this performance always makes me tear up not only because she does it beautifully but because this was the first episode of Letterman after 9/11. It's so heartfelt that I'm pretty sure Dave's trying not to cry at the end when he hugs her. It's beautiful.
And I'll throw this one in because while this is a spoken word version, there's also a song version.
Leonard Cohen's "A Thousand Kisses Deep." I like it done as spoken word much better than the song version, it's fantastic. I got to hear him do this at the show I was lucky enough to attend, and I think I held my breath the whole time.
"My mirror twin, my next of kin, I'd know you in my sleep. And who but you would take me in, a thousand kisses deep?"