The original question is something that my musician friends and I talked about from time to time (How Many Times Do You Have to Listen to Understand a Piece of Music?). At first exposure it's not so much about understanding the piece but rather really liking the piece. To illustrate the point, do you hear the lyrics or the music first? My wife hears the lyrics first. I (along with several musician friends) hear the music first. In fact, I hear the music (the beauty of chord progressions, etc.) and the first three or more times I hear it. The meaning of the lyrics comes later (and I do love great lyrics).
As far as understanding the piece, that comes within the first couple of listens as well. If I'm learning to play or perform the song, the understanding becomes very deep. And it will be listened to dozens of times to truly climb into the orbit of the song and represent it well.