I'll throw out the easy one: 1876.
By Bryce Johannes, Explorer of Lesser Known Histories in Search of a Brighter Future
Unrestricted rule by Republicans since the Civil War had let corruption and favoritism to the well-connected become the order of the day. The government had become a way for the wealthy elite to reward people loyal to them. Since positions were rewards those who filled them looked for ways to make the most (profit) out of the position.
The Republicans were so confident that the Democratic reputation was beyond recovery that they ran openly corrupt politicians. It was in this context that in 1876, the Democrats actually managed to take a majority of the vote.
The results of the election were in doubt because of election fraud by both parties (including violence and intimidation throughout the South that disenfranchised many African-Americans). The Democrats eventually accepted the election of Rutherford B Hayes in exchange for an end to government interference in human rights violations in the South (typically sanitized as the "end of Reconstruction").
The Constitution originally provided for the person who received the second-fewest votes for president to be the vice president, the most to be president (electors had 2 votes). With the onset of political parties, however, this became problematic, and in 1800, Thomas Jefferson and Aaron Burr received an identical number of votes. The race was thrown into the House of Representatives, which elected Jefferson. This led to the 12th Amendment, which led to separate elections for president and vice president.
Four candidates ran for president, with the result that no one received a majority of the popular vote. This threw the election to the House, which elected John Quincy Adams. The supporters of Andrew Jackson, who had won a plurality of the popular vote, were furious, and derided Adams' election as the result of a "Corrupt Bargain." Four years later, Jackson stormed back and captured the presidency, becoming the first Democratic president.
Already covered. Worth noting: Tilden won the popular vote by three points, although suppression of the black vote in the South doubtless contributed to this.
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