Ha! This is the closest thing we have to “our song”. I picked it for our wedding dance because it was the only slow song we both liked. I bailed out halfway through the dance because I couldn’t stand people looking at us any longer. Now that memory makes me cringe in embarrassment, and I bet you any amount of money my husband couldn’t even tell you what song was played.
Journey’s Faithfully because “…livin’ with a music man ain’t always what it’s supposed to be.”
Yeah, I am just a director for a technology company, but for some reason that was just “our song” even though it really didn’t fit us at all. However, we are married 25 years in July, so thank you, Steve Perry. 
In terms of romantic songs, it would be “Shining Star” (by the Manhattans, not the EW&F song). We spent some time thinking about what we wanted to be played at our wedding, and after narrowing down a few of our favorites, we settled on that one. Which, of course, the chapel forgot to play after the ceremony <grumble grumble not that I’m bitter grumble grumble>. Still our song, though. 
But as an overall song, I’m not sure any song has brought us more joy and silliness during the time we’ve known each other than the timeless “Dragostea Din Tei.”
Boom King by Flight of the Conchordes. Still makes us laugh. Laughter is a big part of what keeps us together. Neither of had laughed in years before we got together, we pretty much haven’t stopped in 5 years.
What?! Our song?! We are supposed to have a song? Oh gawd the pressure. I’m really not sure what it would be if we had one…maybe one of his own songs. He is a song writer and musician. I love to listen to him play.
It has been fun reading the comments.
This would be the closest:
https://www.youtube.com/watch?v=hzRmp-O496Y
We don’t really have a romantic one 
Our first wedding dance was, of course, “Could I Have This Dance For The Rest Of My Life” by the incomparable Anne Murray.
Things have changed a bit. Now it’s a Jimmy Buffet love song…
If my husband picked one today it would be Cheeseburger in Paradise.
Ours is one that is a bit weird for a couple falling in love…
It is Olvidala by the Colombian Vallenato group Binomio de Oro. I was the one who really liked it and Sra Iggy thought it was rather inappropriate. My Spanish is much better now and I understand where she was coming from.
[spoiler]
The song is a duet between two males. One is telling his lovelorn buddy to forget her, after all she left you because you had another love.
While his the lovelorn friend laments he cannot forget her. She was his first love.
So… cheating guy gets all stalker-ish when his first love leaves him.[/spoiler] Yeah… great couples song for Sra Iggy and I. :dubious:
At Last, by Etta James, or by Jimmy Scott, or by Ruby Wilson, the Queen of Beale Street.
Once a friend asked The Squeeze and I if we had a “song.” We were kind of inebriated at the time and answered, I swear, in unison, Why Don’t We Get Drunk And Screw?. Hence, our song!