There's a long-lived theory that Tyrion, is a secret member of House Targaryen. In the core of this theory are speculations on that the late Mad King Aerys is his father, who raped Tyrion's mother Joanna Lannister during her final visit to King's Landing. If true, it gives literal meaning to Tywin's revelation that Tyrion is not his son.
Another one proof of this theory we can find in the in-universe history book written by George R.R. martin “The World of Ice and Fire”. In the chronicles chapter, the Mad King pays close attention to Aerys' obsession with Tyrion's late mother, and the resulting tension it created in his relationship with Tywin.
If it turns out that Tyrion is indeed a Targaryen, then it's worth looking at his relationship with Daenerys' dragons in a different light, and the third dragon head.
Prediction and article by Vahe of the House Lannister of Casterly Rock