Last dragon

The last dragon ​belonging to House Targaryen died many years before the events depicted in Game of Thrones. It's death marked the extinction of dragons in Westeros and beyond until the birth of Daenerys Targaryen's dragons.

Biography
​At Vaes Dothrak, Viserys Targaryen bathes with Doreah and she tells him that she wants to see a dragon more than anything and asks if he has seen one. He replies that he has not, as the last one died many years before he was born. He tells her that he saw their skulls when he was a child, as the were used to decorate the Red Keep. He tells her that the ones closest to the door were the last ones they were able to hatch. He says that they were stunted and wrong, their skulls no bigger than a dog's.

Petyr Baelish believes that the last dragon's death was a precursor of the eventual downfall of the Targaryens.

In the books
In the A Song of Ice and Fire, the last dragon was a green female. She was sickly, small, misshapen and stunted, with withered wings. Her name is not known. Although she was sickly she did lay a clutch of five eggs, which never hatched.

In The Hedge Knight, the first of the "Tales of Dunk and Egg" prequel novellas, Ser Duncan the Tall remembers his late mentor Ser Arlan of Pennytree telling him that he had seen the last dragon when he was a child.

When Tyrion Lannister views the nineteen dragon skulls stored in a dank cellar in King's Landing he recalls seeing a matched pair of skulls, no bigger than mastiff’s skulls, and oddly misshapen, all that remained of the last two hatchlings born on Dragonstone. He thinks to himself that they were the last of the Targaryen dragons, perhaps the last dragons anywhere, and they had not lived very long. He remembers telling his uncle Gerion as a child that he wanted a pet dragon, but Gerion only laughed and told him that the last one had died over a century ago.