In 1980, Jimmy Carter became the first since 1932 to lose his re-election bid.
In 1976, Gerald Ford became the first incumbent to lose since 1932. However, he was not running for re-election since he was not elected the first time.
Lyndon Johnson chose not to run in 1968 and so avoided the likelihood of losing a reelection bid.
George H. W. Bush lost in the election for a second term against Bill Clinton in 1993. Back in the 1970s, Richard Nixon was elected, but did not serve his full second term; Gerald Ford was not elected after serving as Nixon's replacement; and Jimmy Carter just served one term.
The president in 1932 was Herbert Hoover. He left office in 1933 and was followed by Franklin D. Roosevelt.
George W. Bush, Sr. President Bush served his first and only term from 1989 to 1993. He lost his reelection bid in 1992 to Bill Clinton.
I am not sure what you are asking. A sitting president is normally certain of being nominated for a second term, so he tries to appear statesman-like and keeps apart from the hoop-la of the convention. If he wants a new Vice President or has ideas for the platform, he would let his staff and friends make his wishes known. He will probably appear and make a speech the last day of the convention.
Spiro Agnew.
George H.W. Bush was defeated by Clinton in 1992.
Theodore Roosevelt
Woodrow Wilson (1919)
I think this person is asking what the last date that a sitting president is in office and if so, it is January 20 of the year following an election that he did not win.
Adams
John Kennedy was elected president after Dwight's second and last term.
As I write this answer in November 2012, the president last year was Barack Obama. He has just won a second term as president, so he will be president next year too.
Roosevelt was in a wheel chair. Does that qualify?
There is no president who meets your criteria.