It's possible that they would go on a mission to uncover some long - lost secrets in the world of 'Game of Thrones'. Jaqen's knowledge of the underworld and Arya's determination could lead them to places like the ancient Valyrian ruins. There, they might find powerful artifacts or knowledge that could change the balance of power in Westeros.
One possible adventure is that they get involved in a heist. Maybe there's a valuable item in a heavily guarded castle, and they plan to steal it. Jaquen could use his skills of disguise and Arya her quick thinking to pull it off. Along the way, they'll face guards, traps, and other obstacles. This adventure would not only be about getting the item but also about their growth as a team.
At first, it could be a teacher - student relationship, with Jaqen teaching Arya more advanced skills of the Faceless Men. As time goes on, they might become comrades, relying on each other in dangerous situations.
Arya may become more confident in her abilities with Jaqen's guidance. She could grow from a somewhat reckless girl into a more calculated and strategic assassin. For example, she might learn to better plan her attacks instead of acting on impulse all the time.
In this fanfic, they might embark on a journey to explore new lands. Arya could learn more about the Faceless Men's ways from Jaquen. Maybe they'll encounter old enemies or make new allies along the way.
In some fanfictions, there might be a plot where Jaqen actually has a deeper connection to Arya's family than she initially realizes. Maybe he was secretly tasked with protecting her all along. This could lead to a series of revelations and a change in their relationship dynamics, from mentor - student to something more like comrades in a grander scheme.
She might have a small, but fast and agile dragon. Since Arya is all about speed and stealth, her dragon could be like that too. It could be a dragon with dark scales, which would blend in well with the night sky when she's on her secret missions.
Another plot could be centered around Jaqen teaching Arya some of his unique skills. Arya is already a skilled assassin, but Jaqen has his own brand of magic - like his ability to change his face. He could pass on some of this knowledge to Arya, and they could use it to infiltrate enemy strongholds and carry out their plans.
One common theme is the growth of Arya. With Jaqen in the story, Arya often has to learn new things, whether it's about combat or about the world around her, so her growth as a character is a big part.
Well, in the world of 'Game of Thrones' fanfiction, the possibilities are almost endless. Since the show and books have such rich lore and a large cast of characters, it's not surprising that someone would write a fanfic where Arya hatches a dragon. This kind of plot would likely involve some sort of magic or unforeseen event that leads to Arya coming across a dragon egg and then having it hatch. It could also potentially change the entire course of the story in the fanfic, as Arya with a dragon would be a very powerful force.
She might get a small but fierce dragon, like a young and agile one. It could be a dragon with black scales, which would match Arya's dark and mysterious nature.