This is predicting WAY into the future...
Ash will continue to get better at Pok?mon battles. He will eventually become a Pok?mon master in his own eyes (meaning, he might not necessarily have caught every existing Pok?mon, but he will have caught and trained enough to feel as if he's accomplished enough), and he will return home. He will be a hero in Pallet town, and young trainers everywhere will recognize him and ask him for tips. He will be a real celebrity at home.
Misty will have taken over the Cerulean Gym. Her sisters will have gotten tired of being gym leaders after so many years, and move on with their lives.
Brock will return home, too. His brothers and sisters will have grown up since then, and now instead of chasing girls 24/7, he will be busy chasing away guys like that remind him of himself from his little sisters. He will go back to being a gym leader/breeder, and by chance meet a girl with the same interests as him, and marry her.
May and Max will return home, and eventually take over at their father's gym.
Professor Oak will retire from Pok?mon-research, handing over everything to Gary. Gary and Ash's rivalry isn't as tense as it used to be, and they now regard each other as friends. They will probably work together to help younger Pok?mon trainers on their journeys. Oh yeah, and since Ash's father is out of the picture, Ms. Ketchum will probably remarry to Prof. Oak, which will result in a really strange family tree.
Jessie and James will eventually have a successful mission, and be promoted in Team Rocket, making Butch and Cassidy look like pathetic losers. However with no 'twerps' to chase after, they will realize that maybe being evil isn't their calling. Luckily, James' parents will probably end up leaving him some inheritance despite what was said in the first season: not much, but enough for him to start a comfortable life for himself. He and Jessie will probably get together, because after everything they've been through together, seeing other people is out of the question. James' growlithe will probably end up reuniting with James, and will serve as a loyal, family pet. Meowth will probably live with them too. And with James' newly-inherited money, he will be able to pay off all their debts to Team Rocket.
Tracey will probably return once or twice as well, just to catch up with his old friends. Maybe he'll get a job at the police station as a sketch artist, and ride around with Officer Jenny on the motorcycle.