Warning: spoilers ahead for the latest episode of Game of Thrones, "The Queen's Justice."
Game of Thrones has been building to the meeting of Jon Snow and Daenerys Targaryen from nearly the very beginning, and fans finally got to see the two face off in "The Queen's Justice." Considering that she was backed up by three dragons, an Unsullied army, a Dothraki horde, and her own heritage as a Targaryen, Dany was feeling pretty confident about her claim on the Seven Kingdoms, and she wasn't a big fan of Jon touting himself as king of one of those Kingdoms. The song of ice and fire began, and ice was at a distinct disadvantage. As it happens, however, their very first conversation may have revealed several reasons why why Dany might never take the Iron Throne for herself.
For one thing, her meeting with Jon -- which began with Missandei spouting off Dany's many titles -- proved that she places a great deal of value on the fact that she is the last Targaryen, and therefore the only rightful claimant to the Iron Throne. Her father is dead, both of her older brothers are dead, and both of Rhaegar's kids with Princess Elia were murdered. As far as Dany is concerned, she is the next (and only) in line to rule. The Targaryen name means very much to her, and she believes her right to the Seven Kingdoms is based on the line of succession. She made this very clear to Jon.
What was not made clear to her is the fact that she's almost certainly not the last of the Targaryen bloodline. Although it has not yet been explicitly confirmed that Rhaegar Targaryen was Jon Snow's father, we can be pretty sure that the reveal will come that Jon Snow is a blood relative of Dany's. We know that the Targaryens were totally cool with multiple wives, and the show has hinted that Rhaegar loved Lyanna Stark, so we can infer that the two were married when Jon was born, making Jon a legitimate Targaryen, even if he considers himself the son of Ned Stark.
The fact of the matter is that Jon is not Ned Stark's son. As the (presumed) son of Rhaegar Targaryen, Jon is technically next in the line of succession of the Targaryen bloodline. If Dany sticks with her claim that the next ruler should come from the next in the Targaryen line, then she should technically bow out and do everything she can to put Jon on the throne. Given that there's no way in seven hells that Dany will give up her claim now, she may find her cause imploding if the truth gets out that Jon would have had the Targaryen name if not for Robert Baratheon's rebellion. She said in "The Queen's Justice" that she got as far as she has due to her unshakable faith in her destiny to rule; can she really succeed if that faith is shaken?
Of course, Jon has proved that he has absolutely no interest in ruling the Seven Kingdoms, so it's possible that his status as a Targaryen won't have much more impact on the story than riding a dragon or something equally symbolic. Still, her meeting with Jon in "The Queen's Justice" was the first time that she met with a Westerosi leader who hadn't fled to her in exile, and the way she conducted the conversation could point to another reason why she might not take the Iron Throne.
For all that Dany doesn't want to take the Seven Kingdoms by simply roasting the countryside, she was quick to play the "I have dragons and I can burn the North to ash" card and claim that Jon owes her his fealty due to a vow from his ancestor a few centuries prior. As Jon said, he doesn't know her, and so he doesn't have any reason to bow to her whims just because she says so. The reign of the Mad King Aerys only ended a generation prior to the Game of Thrones action, and we can bet that the North isn't the only one of the Seven Kingdoms that remembers the destruction of the last Targaryen ruler.
If Dany demands fealty of other Westerosi lords in the same way she did Jon, she might find herself facing a rebellion rather than a kingdom of willing subjects. Yes, she has dragons and armies, but Euron Greyjoy has already seriously crippled her war effort with his fleet, and Tyrion made the valid point that it's much to risky to send Dany herself across the ocean on the back of a dragon to try and find Euron's fleet to burn. Her ability to survive flames doesn't mean that she'll be able to survive arrows or swords or knives. While her ancestor did take Westeros with three dragons a few centuries ago, he wasn't the only Targaryen involved in the conquest, and a lot has changed in the last 300 years of Westerosi history.
There's also the point that Dany flatly refused to believe that she could possibly need Jon and the North for anything, what with her dragons and Dothraki and Unsullied. Given how quickly she has lost so many of her resources in Season 7, she's going to need allies soon, and her position for bargaining with Jon won't be nearly as strong as it was when they first met. There are more players who are more accustomed to scheming in Westeros than she had to deal with in Essos, and burning bridges won't make her path to the Iron Throne any easier. Her deal to give Jon access to the dragonglass at least shows that she can be talked into compromise.
Finally, Dany never seemed more truly Targaryen than she did in the big meeting with Jon, and I don't quite mean that as a compliment. We've only seen a few other Targaryens on Game of Thrones. Maester Aemon was a good egg at the Wall, but Aerys (via Bran vision) and Viserys didn't exactly have all their marbles, and they were both supremely confident until almost the very end. Aerys' last words were "Burn them all," and Dany was more or less threatening the same thing to Jon and the North if he didn't bend the knee. Could we have a Mad Queen Daenerys in the not-too-distant future?
After all, Dany is the product of far more generations of incest than any of Cersei and Jaime's kids, and some of her ordeals have been horrific enough to drive even a sane person half crazy. Rhaegar sounds like the sanest of the Targaryen bunch, and he was still enough off his rocker that he thought it was a good idea to elope with the daughter of a Northern lord who was betrothed to the lord of House Baratheon. It's quite possible that somebody in the Seven Kingdoms will be concerned at the possibility of a new mad Targaryen ruler and try to stop her. Maybe a Queenslayer will appear at some point.
All of this said, there are still a ton of variables in play, and it's likely that Dany will learn some key lessons from the events of "The Queen's Justice." As much as I'd love to see Emilia Clarke crush a Mad Queen storyline -- and I believe that there have been enough clues dropped about potential developing madness for the storyline to work --- she could continue to push forward as the Breaker of Chains and Mother of Dragons who only wants the best. She could still take the Iron Throne. There's no current candidate who's especially good for the job at the moment, and we may discover that she's the best option for the commonfolk, mad or not.