If you count the pieces I play in orchestra as "songs" and as "annoying", then I would have to say Serenade for Strings movement IV by Tchaikovsky is on my list of annoying songs that's always stuck in my head. XD
I guess, technically I get more tunes and melodies stuck in my head than I do the actual lyrics of songs, if that makes any sense. I could totally be humming something and not think about the lyrics, until someone goes up to me and says, "Hey, isn't that Every Time we Touch by Cascada?" and I would go all "Wha? Is it?" because I'm totally ignorant like that.
But really, there's Danjo that somehow keeps finding its way into my mind, along with "hot cross buns" for some odd reason.