Yes, there might be some good ones. You can try searching on some popular fanfiction websites like Archive of Our Own. Just make sure to check the ratings and reviews to find well - written and appropriate stories.
One key element is the relationship dynamic. Since Drogon is a dragon and Jon is a human, exploring how they interact, communicate (if there is any form of communication in the fic), and form a connection is central.
There is a fanfic called 'Drogon's Bond with Jon' where the author delves deep into how Drogon, being a powerful dragon, and Jon, with his own complex backstory, form an unexpected alliance. It's a great read if you like stories that focus on the bond between different beings. In this fanfic, the author vividly describes their interactions, from the first encounter which is filled with suspicion to the later stages where they start to rely on each other.
Perhaps Jon Snow and Drogon meet in a desolate place. Jon is alone and injured, and Drogon, instead of attacking, shows a strange compassion. Jon, using his wits, figures out how to communicate with Drogon. They then form an unlikely bond, and Drogon helps Jon get back to civilization. This would be a great character - development - based fanfiction scenario.
Yes, there are quite a few. Some explore Jon's new identity within the context of the Targaryen legacy, delving into how he might handle the power and politics associated with it. For example, there are stories where he reunites with Daenerys and they have to navigate their relationship in light of their shared heritage.
Their relationship could develop into a very strong father - son bond. Rhaegar, having lost so much, could pour all his hopes and dreams into Jon. Jon, on the other hand, could look up to Rhaegar as the father he never had. They might travel together, facing various challenges and growing closer through shared experiences, like fighting off enemies or dealing with political intrigue in Westeros.
In the fanfiction world, Jon Snow meets Drogon and there could be an instant connection. Since Jon is a Targaryen (in the show's revelation), Drogon might sense a kinship. They could form an alliance. Jon could use Drogon to gain an upper hand in battles or to travel quickly across Westeros. It would be an amazing sight to see Jon Snow on the back of Drogon, flying over the kingdoms.
Well, a recurring theme is the revelation of his true heritage and how it impacts the overall plot of the 'Game of Thrones' universe. There's also the theme of loyalty. Does he remain loyal to the North or does his Targaryen side pull him towards other allegiances? And family is another theme. How does his newfound Targaryen family change his view of family compared to his Stark upbringing?
One idea could be that they meet during a political intrigue in King's Landing. Female Jon Snow, with her Stark-like honor, might initially be suspicious of Aegon Targaryen. But as they get to know each other, they find common ground in their desire to bring peace to the Seven Kingdoms. They could form an alliance that eventually turns into a deep friendship or even more, all while facing various threats from other power - hungry lords.