I may not understand your question but Gerald Ford did seek the presidency in 1976. He was nominated by the Republicans but lost the election to Jimmy Carter.
Pat Buchanan sought the Presidency, but was never elected President.
Millard Fillmore became the 13th President of the United States, succeeding Zachary Taylor, who died in office on July 10, 1850. In the Election of 1854, Millard Fillmore sought, but was not nominated, the Whig candidacy.
She was Republic party nominee John McCain's vice-presidential candidate in the 2008 election. She has not sought election as president, though many speculate that she may in the future.
Emperor Nero himself sought to rebuild the Capital.
President Gerald Ford was never elected to President. He assumed the position when Richard Nixon resigned over the Watergate scandal.
I am not sure that he really had any poiltical philosophy when he first sought the presidency- I think it was a more a matter of keeping Clay, Adams and other enemies out. Bur eventually , he became a believer in a more direct form of democracy than was set by the constitution. He advocated the direct election of senators and even of supreme court justices, and for the election of the president. He also wanted term limits and shorter terms for some offices.
James Polk was the first president who did not seek re-election after serving his first term in office. Other presidents before him did not serve two terms, but they sought re-election and failed to get it.
Although he sought to endear himself to me I found him to be repulsive and fawning.
"silent majority", a term he used to refer to conservative Americans who were believed to be largely supportive of his policies but were not vocal about it. Nixon sought to tap into the frustrations of this group and position himself as their representative, promising to restore law and order and bring an end to the Vietnam War. This appeal helped him secure a narrow victory in the election.
Reagan was a conservative member of the GOP. Basically, his presidency focused on the economy and expanding the military. During is presidency he also sought to, and eventually succeeded in, ending the Cold War.
Initially Lyndon Johnson sought re-election but he realized that that was not popular so he had Hubert Humphrey run. Bobby Kennedy got into the race late. Richard Nixon was the Republican candidate.
The existing holder of an office. For example, in the 2004 United States presidential election, George W. Bush was the incumbent, because he was the president in the current term while the election sought to determine the president for the following term.