Not in the Harry Potter books one through seven. The last book ends with his final year at Hogwarts in which he can't become Headmaster because he has learning to do. If the author made another book about what happened after Harry finished Hogwarts, it is possible he could take over the job as Headmaster.
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No. Harry never went back to Hogwarts for his seventh year because he had to destroy the horcruxes and he never went back the year after because he didn't need to and Harry isn't the type of person to sign up for more school.
technically he doesnt ever graduate at all, at least there is no direct mentioning of it in the book
i No, Harry does not graduate he stops going to Hogwarts in his 6th year to hunt down crucifixes to destroy Voldemort. (I mean, "The One Who Shall Not Be Named")
Harry Potter never graduated from Hogwarts. He dropped out and never returned to finish up his schooling.
Hogwarts exists only in the Harry Potter series.
He spent it at Hogwarts.
He is the caretaker.
Harry says that Hogwarts would not be the same without Dumbledore. He also wants to finish the search for horcruxes to defeat Voldemort. However, he does return to Hogwarts at the end. If he did return he would be killed by Voldemort who was running the school.
He left at the end of Harry Potter and the Prisoner of Azkaban.