I'd never heard of this questionable claim, but apparently some people believe that David Rice Atchison actually was President of the United States for one day.
Whatever his other accomplishments as a lawyer and statesman, David Rice Atchison will always be known as the guy who was President of the United States for just one day.
He served as a US senator from 1843 to 1855. As president pro tem of the Senate in 1849, Atchison was shoved into the highest office in the land in a constitutional cusp between James K. Polk and Zachary Taylor, because Polk's term ended on a Sunday at noon, and Taylor's inauguration wasn't until the next day.
Wikipedia attempts to debunk this claim, but I don't think it's successful.
While it is true that the offices of President and Vice President were vacant, Atchison in fact was not next in line. While the terms of James K. Polk and Vice President George Mifflin Dallas expired at noon on March 4, Atchison's tenure as President Pro Tempore did as well. He also never took the oath of office, although there is no constitutional requirement, then or now, for an Acting President to do so. No disability or lack of qualification prevented Taylor and Fillmore from taking office, and as they had been duly certified as President-elect and Vice-President elect, if Taylor was not President because he had not been sworn in as such, then Atchison, who hadn't been sworn in either, certainly wasn't.
That's somewhat confusing, but it appears that Wikipedia argument is that Atchison didn't take an oath, either to remain Senate President Pro Tempore or to assume the Presidency, and that if an oath was required for Taylor to be president, then an oath would have been required for Atchison as well. The Constitution does require that the President take an oath of office (Article II, Section 1):
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:--``I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.''
However, before the 25th Amendment was passed, a preceding paragraph allowed that "the Congress may by Law provide for the Case of Removal, Death,
Resignation or Inability, both of the President and Vice President", and at the time the law provided that the Senate President Pro Tempore was next in line; the law didn't mention an oath requirement. Atchison's term as President Pro Tempore expired at the same time as Polk's presidency, but seeing as how he immediately continued in the office it seems logical to conclude that there was no legal break; he didn't take the oath as Senate Pro Tempore again until the next day, but the Constitution doesn't require an oath for that office. Therefore, Atchison's oath deficiency need not have prevented him from being Senate President Pro Tempore, and consequently President.
Anyway, it appears that even if Mr. Atchison was president for a day, he spent it asleep in bed.
Snopes has more reasons Atchison shouldn't be considered to have been president, and they're all basically as convincing as you want them to be. They also make the mistake, though, of assuming that a person becoming president based on the provision of Congress would have to take an oath. However, they do rightfully point out that the Constitution only says that the person selected by law only acts as president, but doesn't necessarily become president.
Furthermore, the US Senate website dismisses the Atchison presidency -- though with an off-handed "of course" that seems unjustified to me.