One possibility is that Percy, raised by Chaos, would have a unique understanding of the balance between order and chaos. He could see the flaws in the current system of the demigod world which is so focused on order. This could lead to him trying to bring about some sort of change, whether the other demigods like it or not. His fighting style might be more fluid and unorthodox, taking inspiration from the chaos that raised him. And his personality could be more aloof, as he doesn't quite fit into the social structures that the other demigods are used to.
Well, perhaps Percy would grow up with a very different set of values and skills. He might be more wild and unpredictable, having been raised by Chaos. Maybe he'd have powers that are more primal and less bound by the normal rules of the demigod world.
One idea could be that Percy has unique powers given to him by Chaos. For example, he might be able to control the very fabric of the universe in small ways. Maybe he can manipulate time on a very limited scale, like slowing it down during a fight to gain an advantage.
The Titans could still be antagonists. Even though Percy is raised by Chaos, the Titans might see him as a threat to their plans of overthrowing the Olympians. They could try to capture or kill him to prevent him from interfering.
One possible plot could be that Percy grows up learning the ways of the hunt. He might become an excellent tracker and fighter under their guidance. For example, he could be trained in archery and stealth, skills that are crucial for the hunt. He may also form strong bonds with the members of the hunt, and they could play a big role in his character development, perhaps making him more in tune with nature.
The plot could involve Percy being raised away from the typical demigod chaos under Hestia's care. He may be trained in the ways of maintaining harmony and peace. Hestia could use her powers to shield Percy from some of the more dangerous aspects of the demigod world, and in turn, Percy might become a sort of ambassador for her values among the other demigods, promoting the importance of a stable home environment and the bonds that come with it.
Maybe in the fanfiction, Percy is on the verge of death in a battle. Chaos, who has been observing Percy for a while, decides to save him and adopts him. Percy then embarks on a journey in Chaos's realm. He could be trained by powerful beings there and form new alliances. His relationships with his old friends from his previous life might be strained as they don't understand his new situation, but he also gains new friends who are loyal to him because of his connection to Chaos.
Well, perhaps Percy could be learning some new and powerful combat skills from Chaos. Maybe Chaos teaches him to manipulate chaos energy in unique ways for battle.
Well, in a 'Percy meets Chaos' fanfiction, it might start with Percy being pulled into a strange dimension where Chaos exists. Chaos could be a formless entity that communicates with Percy telepathically. Percy has to learn to understand Chaos' motives, which could be to reshape the world Percy knows in a more chaotic way. Percy then has to decide if he wants to fight against Chaos or join it in some way.
In such a fanfiction, Percy could have a very different personality. Since Amphitrite is a goddess, she might raise Percy with a more formal and disciplined upbringing compared to his normal life. He could be more in tune with the politics and hierarchies of the underwater world, and perhaps have a more regal bearing. He might also have a special relationship with sea creatures that are more closely associated with Amphitrite, like certain types of sea nymphs or powerful sea monsters.
The fanfiction might explore Percy's relationships in a new light. For example, his relationship with Annabeth could be strained as he now has loyalties to Gaea. His friendships with other demigods might also change drastically. Maybe he'd be seen as a threat and have to fight against his former allies. Overall, it would be a complete upheaval of the original Percy Jackson story dynamics.
One possibility is that Percy grows up with a strong sense of independence and a connection to nature. Since Artemis is the goddess of the hunt, Percy might be trained in hunting and survival skills from a young age. He could also have a special bond with the animals in the forest, being able to communicate with them to some extent. This upbringing would likely make him a formidable fighter, not just with a sword but also with bows and arrows and stealthy combat techniques.