Nine Presidents never attended college: Washington, Jackson, Van Buren, Taylor, Fillmore, Lincoln, A. Johnson, Cleveland, and Truman. The college that has the most presidents as alumni (six in total) is Harvard: J. Adams, J. Q. Adams, T. Roosevelt, F. Roosevelt, Kennedy, and G. W. Bush (business school). Yale is a close second, with five presidents as alumni: Taft, Ford (law school), G.H.W. Bush, Clinton (law school), and G. W. Bush.
according to: http://www.infoplease.com/spot/prestrivia1.html
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Harvard University has produced more US presidents (7) than any other university. (http://en.wikipedia.org/wiki/List_of_United_States_Presidents_by_college_education)
More US presidents were lawyers before they became President. 25 of the 43 presidents were admitted to the bar, although not all them earned their living that way.Of course, if you consider politician to be a profession, even more of them engaged in that.
Harvard University, located in Cambridge, Massachusetts, has produced the most U.S. Presidents, which is eight. Harvard also produced four US Vice Presidents.
George Washington is the president that appointed the most Supreme Court Justices. There have been only seven US Presidents that served only one term.
No US presidents were named Nathaniel.
If you're asking whether any US Presidents were constitutionally ineligible to hold office, the answer is no.