800s C.E. Agrabah.
Author's note: This hinges on the premise that Abis Mal did not find the lamp and therefore Jafar wasn't killed, in other words, the sequel did not happen. Aladdin and Jasmine still got married and many of the enemies of Agrabah still exist, not because I like them but because their existence doesn't FUBAR my premise. Also, there are two reasons Genie can't find the lamp. One is the prophecy, and one is that genies cannot counteract more powerful genies. Jafar is more powerful than Genie since he is not free, and Jafar could easily have made it so that Genie or Aladdin simply cannot find his lamp.
"What's on your mind, Jasmine?"
"Nobody's found that lamp yet. I still think we should have destroyed it. What if someone finds it and helps Jafar get revenge on us? Not to mention it could be anywhere. It could be in another kingdom. What if our enemies find it? Can you picture Mozenrath finding that lamp? I don't want to."
"We'll just have Genie find it."
"Um, Aladdin, when I threw it I still had phenomenal cosmic powers. Now I'm only semi-phenomenal, nearly-cosmic. That meanie genie in the black lamp is still phenomenal and cosmic."
"Well, you can still find the lamp, can't you?"
"I can try."
"And we're going to consult the new Vizier," said Jasmine. While Aladdin had turned down the position, a cousin of Jasmine's had stepped up both to advise Sultan Hamed and to train Aladdin and Jasmine in being good rulers.
"Omar, what would you do if, hypothetically, your predecessor was a very bad man, a usurper, who turned into a genie but three years later the lamp still wasn't found?"
"That's a little specific for a hypothetical."
"Yeah, it really happened. Sorry."
"Well, while we were looking for the lamp, I'd warn the populace and the rulers of the surrounding territories."
"Baghdad and Samarra will know what to do."
"You can't rely on them."
"Well no, but they can help us."
"We should write a proclamation telling everyone to not handle oil lamps they randomly find in the desert."
"If we tell them that they'll do it."
"Your grasp of the human mind is stunning for one so young. What do you propose we do?"
"Tell them all the truth," said Aladdin. "I learned three years ago that you can't hide the truth for long or it comes back to bite you."
"Yes. If we explain why they shouldn't rub the lamp, they won't. But if we make it sound like an arbitrary instruction, they'll practically be looking for it."
Within a few days, signs had gone up, a proclamation had been read in the public squares, and letters had been written to other rulers.
"Just one more thing to do," said Omar. "Add this to the History of Agrabah, so that even those who come after us will know of Aladdin's valiant deeds and the danger still lurking in the desert."
"Good thinking, Omar. I'm glad you're a good vizier."
"One in a thousand."
"Do you think it's a good idea to post guards in the desert so that if someone does find the lamp we can stop them from making a wish?"
"Jasmine, you need to stop asking for advice and start taking action. We always used to."
"Yes. It's just that since Father's been sick I've been thinking about what it takes to run a kingdom. Running off to solve the problem with action doesn't always work."
"I don't always do that, either."
"No, but you are impulsive. Need I remind you of the fruit-juggling trick?"
"The Caliph didn't mind! Did he?"
"Well, we haven't been invited to Samarra, Aladdin, and I think they'd have kept us out of Baghdad if they could."
"So, Jasmine, when we're in charge, we can still have fun, right? I mean, your father always had fun."
"He was so distracted by fun he nearly let this place be destroyed by the one person who should have been looking out for it. I love Father, Aladdin, but I'm not blind to his flaws. We're going to have to do a lot of work. But let's start by dealing with this lamp problem."
"I knew I should have destroyed it myself, but I still wasn't sure killing him would be right."
"He was more than ready to kill you, Aladdin," said Omar. "Jafar would have eaten kittens if it would have helped him. He, more than anyone else in recent memory, deserved death, and if you believe nobody does, that's a weakness. It will never get any easier to send the evildoers to their afterlife, but you'll have to realize that letting them live only causes more pain and death. Think of the city first."
"From now on, I will, but that doesn't help me solve what went wrong three years ago. I didn't even try to destroy the lamp. Genie just grabbed it."
"Your genie is a trusted friend. But he lacks skills," said Omar.
"What's that supposed to mean?" asked Genie, who'd conjured up a chalk board and was doing algebraic equations with one hand while juggling with two others and standing on one foot.
"Charming, very charming. I simply meant you and Aladdin's palace staff do not have-or need-identical skills. I would never say anything against your character or your wonderful friendship with Aladdin."
"Thank you, Omar."
"You're welcome. And you're right. This is no time for idle chatter. Genie, I want you to go looking for that lamp."
"Aww, do I have to? I don't like Jafar!"
"Nobody does, Genie, but you have the best chance of finding the lamp," said Aladdin.
Alas, it was not to be. The realm prospered under Aladdin and Jasmine's leadership for many years, but the evil genie's lamp was never found. Guards wandered the desert and Aladdin's Genie went searching periodically, with no luck. He even went to see the prophet Fashir, only to receive "it was not for you to find" and a cryptic prophecy that he never even told Aladdin because he couldn't figure it out. He told Omar, of course, so that it could be written into the History of Agrabah (even though it made no sense).
Omar became obsessed with maintaining and distributing the History of Agrabah, traveling as far as Medina to warn people. Aladdin never minded, because it was definitely a more benign madness than was usually contracted by viziers.
But the fact remained that while Aladdin and Jasmine left a prosperous kingdom, a well-known history and a beautiful family behind, they lived and, finally, died not knowing what had happened to Jafar's lamp other than that it lay buried in the sand somewhere, and sand was everywhere.