The last surviving Targaryen doesn't even know he's a Targaryen until very late in "Game of Thrones." Jon Snow is raised believing that he is Ned Stark's son, born out of an affair with a woman named Wylla. In reality, he is Aegon Targaryen, the son of Rhaegar Targaryen and Ned's sister Lyanna, making him the true heir to the Iron Throne.
After spending years with the Night's Watch, Jon is eventually hailed as King in the North. He reunites with his surviving Stark "siblings" and takes on the task of liaising with Daenerys Targaryen in hopes of getting her help against the Night King. When he meets her dragons, he is surprised that one of them, Rhaegal, lets him ride him, and he later rides Rhaegal into battle.
Jon and Dany fall in love, unaware that they are actually aunt and nephew. When the revelation reaches them, Dany is concerned that Jon will want to take the throne, while Jon is more startled by the truth of their biological relationship. In the end, Jon stays loyal to Dany until the very end, when he is horrified by her burning of King's Landing. He kills her in hopes of saving Westeros and returns to exile with the Night's Watch as punishment for her death.