The reason Game of Thrones, both the book and the first series had people across the world hooked was not dragons. It was because an extremely intriguing cast of characters, with dynamic backgrounds, clear and concise motivations (or mysterious and elusive ones) played off of each other in dialogue and in contrast.
The Targaryen dynasty is at the absolute apex of its power, with more than 10 dragons under their yoke. Most empires crumble from such heights. In the case of the Targaryens, their slow fall begins when King Viserys breaks with a century of tradition by naming his daughter Rhaenyra heir to the Iron Throne. But when Viserys later fathers a son, the court is shocked when Rhaenyra retains her status as his heir, and seeds of division sow friction across the realm.