Daenerys Targaryen's dragons make her a formidable enemy on Game of Thrones. No matter how massive the forces opposing her are, she still has three dragons on hand who could turn an attack into a barbecue in a matter of minutes. Defeating her might seem like an insurmountable task, but that doesn't mean Cersei isn't going to try – and with the aid of Qyburn's gigantic new crossbow, she might have a chance. But can you kill a dragon on Game of Thrones?
Dragons aren't immortal, so they can certainly be killed. It just isn't easy to take one down. They're truly massive in size (Dany's dragons are comparable in scale to a 747 when they arrive at Dragonstone) and they're covered in tough scales. They also have the handy ability to fly away from danger if they choose. When left to their own devices, dragons can live for centuries before passing away of old age. But there are instances of dragons being killed, so it can most definitely be done.
In fact, Cersei's Qyburn-approved method could work – provided their soldiers' aim is on point and they're able to take out one dragon while avoiding the fiery breath of the other two.
There are a few methods that can be used to kill a dragon, though circumstances would have to be perfect for them to work. Qyburn based his crossbow theory on the fact that Dany's dragon Drogon was wounded by spears while in Meereen. That makes sense, and the crossbow arrow did shatter a dragon skull, but you'd have to hit the dragon in the exact right spot for it to be fatal. That would mean piercing one of their eyes, which is a pretty small bullseye on a giant, scaled, fire-breathing, flying target.
Captivity can also lead to a dragon's death, whether from starvation or the fact that being chained would make them easy targets for those looking to kill them. Fighting fire with fire works too, as in: you can kill a dragon with another dragon. George R. R. Martin's A Song of Ice and Fire details a civil war fought hundreds of years ago during Targaryen rule in which dragon fought against dragon, resulting in the death of many. This seems like an unlikely route because Dany currently has the only living dragons and they wouldn't turn on each other.
Unless a magical horn from the books makes an appearance on the show as well: called Dragonbinder, the horn has the ability to control dragons. Euron claims to have it in his possession in the books, but he's made no mention of it on the show yet. If he does have Dragonbinder, then he could use it to take Dany's dragons from her or set dragon against dragon.
Dragons are incredibly difficult to defeat and it would take more than a few things falling into place for it to happen, but it's not impossible.