(Spoilers extended) The third head of the dragon may not be who we think it is...

Other than Aemon, whose nonpresence in the show only reinforces the strong possibility that he is a red herring, there are only three characters still living with Targaryen blood. Dany, Jon... and Gendry. He is the last remaining Baratheon (even if he is a bastard), a character we've known since almost the beginning of the story, and through his father the closest legal heir to the throne after Jon and Dany.

As the story seems to be trundling towards the theme of Westeros reuniting to stand against the threat of the White Walkers, and with Dany and Jon symbolically ending their fathers' conflict through their partnership, I could easily see Gendry doing the same, becoming an essential member of the Targ-Team after his ranging North, and gaining favor with the dragons by right of his blood.