1868 (The Impeachment)
Ironically, the closest Presidential election in history wasn't actually an "election." In 1868, President Andrew Johnson was impeached by the House of Representatives for "high crimes and misdemeanors." While most Senators believed Johnson was guilty of at least one of the charges and should be removed from office, they also saw one major problem: the ascension to the Presidency of Senator Benjamin Wade. Enough Senators were fearful of a radical assuming the Presidency that the chamber voted to acquit Johnson and Wade lost out by one vote.