Drain You by Nirvana was an "our song" that I shared with my ex. It's the perfect "our song". And the song by the Pixies, it's possibly called "Chained" but I'm not sure, but it goes, "we're chaaaiiined, we're chai-ained, we're chai-ai-aaiiined" ... well, like that, over and over, and it's AMAZING.
Neither of those are the "our song" for my current relationship but they are so filled with emotion without being sooo fuuuccckiiiiinnngggggggggggg cliched that they are pretty much the perfect songs ever in the whole world for couples. Ever.