Season 7 of Game of Thrones has been all about payoff. After tiptoeing around Jon Snow's parentage for six seasons, the revelation that he is half-Targaryen is making viewers look at Jon Snow in a different way. If he had landed on Dragonstone as a Stark, fans likely would've reacted to the wonder and confusion that Jon Snow felt seeing dragons for the first time. Because of his lineage, however, the only way to react to that scene is to ask, "When is Jon Snow going to ride a dragon on Game of Thrones?"
When Jon Snow went to meet with Daenerys Targaryen, he was aware that she had dragons. Despite that, Jon was still in awe of the flying beasts when he landed on Dragonstone. This is mostly due to the fact that dragons are giant flying lizards and there is no way that you wouldn't freak out upon seeing them. Tyrion Lannister even admitted that you "never really get used to them." While Jon may feel foreign to the them, fans know that being a Targaryen gives him a special relationship to the dragons that could prove incredibly useful in the coming war against the dead. Jon Snow is one of the few living people in Westeros who could gain the trust of a dragon, saddle up, and fly into battle.
Dany believes herself to be the last Targaryen, and thus the only person in Westeros, Essos, or the known world capable of raising, training, and commanding dragons. What Dany doesn't know is that Jon Snow's blood may open himself up to the same skills. When discussing a retaliation against Cersei and Euron Greyjoy for destroying a great deal of her naval fleet, Dany suggests riding in on Drogon and raining dragonfire upon them. Her counsel suggests that she not, as one arrow to her heart could put a stop to everything they've worked for. But what if Daenerys didn't have to ride the dragons? What if Jon Snow could?
While Jon Snow probably wouldn't look nearly as cool as Dany while riding dragons into battle, putting Jon Snow on the back of a dragon could solve a major problem for Dany's counsel. This could allow Jon to lead the dragons to battle while Dany and her army are focused on defending Dragonstone or attacking a Lannister army somewhere else in the world. Of course, the largest obstacles here are convincing Daenerys to separate from her dragons, and the fact that no one other than Bran knows that Jon Snow is a Targaryen.
It seems that even though there are road blocks standing between Jon Snow and Dany's dragons, the dragons themselves seem to want Jon Snow to climb upon them. When Jon landed at Dragonstone, they seemed to be especially playful and constantly trying to get his attention. Perhaps if she was able to listen to her dragons, Dany would learn that Jon Snow is truly her nephew and that renting him a dragon could have some major benefits for everyone involved. Until then, fans will just have to settle for Jon Snow sailing back to Winterfell instead of rolling up on a dragon like he clearly deserves to.