President Grant (18th)

Major scandals made Grant's one of the most corrupt administrations. First, his sister's husband tried to aid Jay Gould and James Fisk in a scheme to corner the gold market by manipulating Presidential policies. They failed, but they did get far enough along that the gold market collapsed, causing "Black Friday" in 1869.
The Whiskey Ring Scandal involved much of his Cabinet, including his personal secretary (who he protected). It was discovered that government officials and employees were pocketing the taxes collected on the sale of whisky. Grant stopped this but protected those close to him.
The Credit Mobilier Scandal involved the VP, Congress, and many other members of government. The Credit Mobilier Company was stealing from the Union Pacific Railroad. When they were about to be exposed, they tried to cover up the scandal by offering stock to government officials and employees at huge discounts.
George Washington (1st)

Washington, through the Jay Treaty, gave Great Britain favored-nation status. Thomas Jefferson repeatedly accused Washington of treason because of the Treaty. Washington was going to refute the charges in his farewell address but ended up simply warning about the dangers of political parties.
William Taft (27th)

Taft's head of the Dept of the Interior was accused of interfering with an investigation into coal mining in Idaho. Taft had the accuser fired and cleared the accused. The press had a field day and, in the 1912 election, the progressive wing of the Republican party joined the Democrats. Wilson eventually won that election but it was a contentious time!
Abraham Lincoln (16th)

Lincoln's Secretary of War at the beginning of his term was Simon Cameron, once accused of being so corrupt that the only thing he would not steal was a red hot stove. In 1862, Lincoln asked for his resignation and appointed him as Minister to Russia, thereby sending him where he could do no further harm to the war effort!
Thomas Jefferson (3rd)

In 1802, Jefferson was charged with having an affair with his slave, Sally Hemings (half sister of his deceased wife) and fathering a child. Jefferson denied the charges and remained as president for another 7 years. They eventually had 6 children, according to (sometimes disputed) DNA tests which most historians now accept.
President Andrew Jackson (7th)

During his campaign for President, it was found that Andy Jackson unknowingly married his wife - Rachel - before she was divorced, making her a bigamist. Her ex-husband finally completed their divorce and they were married again, legally, in 1794. But 30 years later, when he ran for President, it became an issue in the campaign. (She died before he took office and Jackson blamed her death on the scandal.) Once he was in office, his Secretary of War, Eaton, was accused of marrying a "tavern maid" whose husband had committed suicide. (High society gossiped that Eaton's affair with her had caused the suicide.) All but one of the Cabinet resigned in protest - their wives led the campaign against Eaton and his wife.
President Hayes (19th) and Mrs Hayes

When Rutherford B. Hayes nominated Stanley Matthew for the Supreme Court near the end of his term, the Senate didn't like Matthew's closeness to several railroads and other corporations. They refused to consider him. The next administration (Garfield) pushed him through but the vote was 24-23, the closest vote for a justice’s nomination ever.
President Warren G. Harding (29th) with wife Florence

The Teapot Dome scandal (before Watergate considered to be the biggest scandal in U.S. history) began when Harding transferred control of naval oil reserves to the Interior Dept. The Secretary of the Interior took bribes from oil companies to disperse the oil. Then there was the Veteran's Bureau scandal, giving cronies jobs . . . A lot went on before he died 2 years into his Presidency.
Chester Arthur (21st)

Chester Arthur was fired from political office as a part of patronage reform by President Hayes. But when Arthur became President (due to the assassination of Garfield), he surprised everyone by running an honest administration. It was said of him: "No duty was neglected in his administration, and no adventurous project alarmed the nation."