Rhaegar Targaryen, Son of The Dragon Gameofthrones

The period after Aegon's Conquest was filled with tragedies for House Targaryen. Lords plotted to dethrone the dragons who took their crowns while the Faith seeks to rebel against the Valyrian practices of House Targaryen. All the while the bonds between siblings deteriorate. Can someone change this fate?
A dragon must have three heads. But, Aegon only had two children. For ill or good, another head joins the dragon.