One possibility is that Harry has to fend for himself on the streets, learning basic survival skills like finding food and shelter. He might meet other outcasts who become his makeshift family, and together they look out for each other. Maybe he discovers some latent magic abilities on his own, without the guidance of Hogwarts, which he uses to protect his new friends from the dangers of the street.
One possible plot could be that Harry discovers his vampire heritage early on. Maybe his parents were involved with vampires in some way. As he grows, he has to hide his vampire tendencies from his Hogwarts friends while also learning to control his new powers, like super strength and enhanced senses. He might struggle with the urge to drink blood and have to find a substitute in the wizarding world.
The fanfic might explore how Harry's upbringing in the wizarding world affects his mental state as he ages. He could struggle with the pressure of being 'the boy who lived' and having so many expectations placed on him. For example, he might have anxiety about always having to be the hero. Also, his relationships with friends like Ron and Hermione could change as he matures and they all face different challenges in life. This could all be part of the psychological growth aspect of the fanfic.
Well, if we consider a 'Harry Potter grows up in Azkaban' fanfic, it could start with Voldemort winning and somehow Harry ends up in Azkaban as a baby. Growing up there, he would have to endure the cold and the constant presence of dark magic. He may learn some unusual magic just from being in that environment, and his relationships with other inmates or guards (if there are any kind interactions) could be really strange and complex.
In a wild fanfic where Harry Potter grows up, he might have all sorts of out - of - the - ordinary adventures. Maybe he'd encounter new magical creatures not in the original books. He could also develop different relationships with his friends. For example, his friendship with Ron and Hermione might take unexpected turns, like having a big fight over a new magical discovery and then reconciling in a more dramatic way.
One possible plot could be that Hermione, having lost her parents at a young age, has to fend for herself on the streets. She might use her intelligence to outwit those who would take advantage of her. For example, she could befriend other street - smart kids and together they form a sort of 'family' where they share resources and knowledge. She could also be constantly on the run from the authorities who want to put her in an orphanage or some less - than - ideal institution. And perhaps, she discovers a hidden magical alley in the city that only she can access, which gives her access to magical resources she uses to survive and learn more magic on her own.
It could be a new dark wizard who targets Harry because of his fame. This wizard might be jealous of Harry's status as the hero who defeated Voldemort and wants to prove himself as more powerful.
In this fanfic, perhaps Harry grows up isolated in the forest and has to rely on his own instincts and the magic within himself. He could stumble upon an old, hidden wizarding settlement in the forest that has been long forgotten. There, he finds ancient spells and artifacts that not only help him grow as a wizard but also give him a unique connection to the past of the wizarding world. This experience in the forest would shape him into a different kind of wizard compared to what we see in the original series.
The plot could develop with Harry having to face the harsh rules of Azkaban. He might start to form alliances with other prisoners who are wrongly accused, like his father's friends perhaps. And as he grows, he starts to understand the dark secrets of the wizarding world from a different perspective, one that is filled with the pain and injustice of Azkaban.
Well, perhaps he could discover new magical abilities related to orgpange. Maybe he'd use these unique powers to fight against the dark forces in a different way than in the original story.