Game of Thrones Season 7 has finally delivered what fans have been waiting years to see: Queen Daenerys Targaryen riding a dragon into battle, putting the entire Lannister army to shame. But now that the dragons are truly in play, there's an enormous target on their scaly backs — every single person in the Red Keep is currently scheming about how to destroy Dany's dragons and quash her rebellion. But can dragons die on Game of Thrones? While it's understandable that Jaime is completely demoralized after seeing Drogon lay waste to the battlefield during Dany's surprise attack and ready to make a deal with Tyrion, the A Song of Ice & Fire books and previous scenes in HBO's adaptation have made it clear that dragons aren't actually immortal, even though they might seem that way.
After all, a well-aimed hit from Cersei's Scorpion bolt (the enormous crossbow operated by Bronn on the battlefield) did wound Drogon. Small arrows are far too weak to pierce the dragons' hide, but with a large enough bolt aimed correctly, the Scorpion could kill a dragon — that's what Cersei was practicing in the dungeons underneath the castle, where the Targaryen dragon skulls are kept. As Qyburn said, dragons are "powerful, but not invincible." He then points out that "if they can be wounded, they can be killed."
According to The World of Ice & Fire, hundreds of years before the events of the show, Dornish warriors killed one dragon, Meraxes, in battle with a very similar Scorpion bolt to the one Cersei uses. But Balerion the Dread, the enormous dragon in the dungeon whose skull Cersei shatters in the above clip? He's the "beast that Aegon rode across the sea," as Qyburn says, before he took over Westeros during the War of Conquest that placed the Targaryens on the throne.
According to the comments section on one of Martin's blog posts, the author says that "[Westerosi king] Viserys I flew Balerion ... When the Black Dread died (of old age, not in war), he did not take a second dragon," and according to the books, Balerion lived for at least two hundred years before dying of natural causes. So not only can dragons be killed with a strategic blow to the eyes, they will eventually die. However, Dany's dragons are very young. The Game of Thrones timeline is a little wonky, especially with all of the condensed travel time in recent seasons, but those dragons are still growing, and they certainly aren't close to hundreds of years old.
The books have contextualized the history of dragons even more thoroughly than the TV show. And the best thing about ASOIF is that George R.R. Martin has added so much background information about the world's history that could be relevant in future installments of the book or TV show. Take, for example, this quote from a Tyrion chapter in A Dance with Dragons:
The eyes were where a dragon was most vulnerable. The eyes, and the brain behind them. Not the underbelly, as certain old tales would have it. The scales there were just as tough as those along a dragon's back and flanks. And not down the gullet either. That was madnes. These would-be dragonslayers might as well try to quench a fire with a spear thrust. "Death comes out of the dragon's mouth," Septon Barth had written in his Unnatural History, "but death does not go in that way."
So that's pretty clear exposition from Martin. In short, a powerful spear through the eye and into the brain, a la Cersei's plan, would work. Everything else? Less effective. Add that you can't drag dozens of Scorpions around to protect ground troops, and there's a very complicated problem for anyone who refuses to bend the knee to Daenerys (except for dragon whisperer Jon Snow, of course). But there is one other way to kill a dragon — another dragon. The Dance of the Dragons was a civil war between two Targaryen factions that forced dragon against dragon.
So Daenerys shouldn't be too confident that she's going to automatically win this war. It's very possible that one or more of her dragons will die, especially one of the smaller ones, who were chained underneath Mereen for a long period of time. But stopping those dragons will be a serious challenge — and whomever attempts to kill dragons on Game of Thrones should have impeccable aim, because without hitting one of those eyes, they're probably out of luck.