As for Jon Snow, he’s not visiting Dragonstone for the purpose of bending the knee or joining Dany’s campaign to take the Seven Kingdoms.
The prospect of dragon fire against the White Walkers has its attractions as well.
Jon wants Dany’s help in preparing for and ultimately fighting the Great War; he’s not especially concerned about what’s happening down in the south with the other six of the Seven Kingdoms.
Thanks to a letter from Sam, he learned that there’s a huge cache of dragonglass at Dragonstone, just waiting to be mined and turned into weapons that can be used against the White Walkers and their army of dead.
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Game of Thrones 7x03 Promo "The Queen’s Justice" (HD)