Well, if Percy leaves camp in fanfiction, it could lead to him exploring the mortal world more deeply. He could end up using his powers in ways that are not restricted by the camp rules. For example, he might start using his water - controlling abilities to help with natural disasters in the mortal cities, which could attract both positive and negative attention. There's also a chance he'll be hunted by monsters more frequently as he doesn't have the protection of the camp's boundaries.
He might go on a solo adventure to discover more about his powers. He could encounter new mythical creatures that he hadn't seen around the camp. Maybe he'll meet some other demigods who are on the run or have their own secret quests.
Well, it could be that Percy leaves camp and discovers a new power within himself while on his journey. He then uses this power in a very public way, like in a big sports event where he showcases superhuman abilities that no one has ever seen before. This could lead to him becoming famous as people are always fascinated by the extraordinary.
He could go to New York City. It has a lot of water sources like the Hudson River where he can use his powers. Also, there are many places to hide from monsters.
Perhaps he'll meet other demigods who have also left Earth for various reasons. They could form a new group and face new challenges together. For example, they might have to deal with a powerful alien being who wants to use their demigod powers for its own evil purposes. Percy would then need to use his wits and strength to lead the group and find a way to defeat this new enemy.
In a 'Percy Jackson leaves earth' fanfic, he might end up in a whole new realm of gods and monsters. Maybe he discovers a parallel universe where the rules of the Greek gods are different. He could encounter new versions of his friends or enemies and have to learn new powers to survive.
In this fanfiction, Percy might use his powers and leadership skills to bring order to the camp. Maybe he reorganizes the training schedules, making them more efficient for the demigods to improve their skills.
Maybe Percy leaves camp to protect his loved ones. He meets someone special during his journey and they have a daughter. He then has to raise her while also dealing with the threats from his past life at the camp.
Well, in this fanfiction, Percy might meet some new characters early at the camp. Maybe he discovers a hidden part of the camp that others don't know about yet.
Perhaps they were caught in a compromising position while having a private moment. This could lead to all sorts of awkwardness among their friends at the camp. They might have to face some teasing or even get in a bit of trouble with the camp counselors for not following the rules.
The plot could involve Percy being abandoned by camp because the camp leaders fear his growing power. He wanders the world, feeling lost and angry. Percy then encounters a mysterious figure who offers him a way to get revenge on the camp. However, Percy, being a hero at heart, realizes that taking revenge won't solve anything. Instead, he decides to find evidence to clear his name and expose the real danger that the camp is ignoring while he was being pushed away.
One possibility could be that Percy leaves camp to go on a solo quest to find a long - lost magical artifact that he believes can save the camp from an upcoming threat. He might face many challenges on his own, like battling new monsters and making difficult decisions without the support of his friends at camp.