Is Tyrion Lannister of Casterly Rock in fact a Targaryen? The mystery deepens...

So the mystery of whether Jon Snow is a Targaryen gets put off for yet another week, as last night's episode of Game of Thrones offered no insight into this mystery. However, being that it's probably true (R+L=J theory), it started to make me think of why this happens to be significant. It has to do with a prophecy in A Song of Ice and Fire and with what Daenerys Targaryen saw in the House of the Undying:

"The fifth room, finally, shows a man very much alike her brother Viserys, except that he is taller and has eyes of dark indigo rather than lilac. He is speaking to a woman who is nursing a newborn babe, telling her that the child's name should be Aegon and saying that "What better name for a king?". The woman asks him if he will make a song for the child, and he replies that he has a song and that "He is the prince that was promised, and his is the song of ice and fire.". He appears to look at Dany then, as if seeing her, and then he adds that "There must be one more," and "The dragon has three heads."

So there you go. Daenerys is obviously one head of the dragon. I think that Jon Snow is the second (and so does everyone else in the world), which leaves the third for us to figure out. My bet's on it being Tyrion Lannister for two reasons: 1) Daenerys's dragons didn't kill him and 2) He's probably a Targaryen too. It looks more and more likely that Tyrion is actually the son of Aerys Targaryen after a rape or seduction of Joanna Lannister. Oy...I think the best part of Game of Thrones is actually trying to figure out what's actually going on in this story as the narrators tend to be a bit unreliable.

So what do you think? Is Tyrion Lannister the third head of the dragon? Is Tyrion Lannister of Casterly Rock in fact a Targaryen?