Two examples of famous people with integrity are Mahatma Gandhi and Abraham Lincoln. Integrity requires that a person only perform actions that square with his inner values. People whose words and actions do not reflect their inner values do not possess integrity.
An anecdote about Gandhi's integrity involves a mother who brought her young boy to meet the great man. She asked Gandhi to tell the boy never to eat sugar because of its effects on his health and teeth. Gandhi refused, instead asking the mother to bring the boy again 30 days later. When she returned, Gandhi did as she wished, telling the boy to stay away from sugar. The mother wondered why Gandhi had not just done that on the first visit, and Gandhi replied that he had still been eating sugar at their first visit and did not feel comfortable telling others to abstain from something that he enjoyed.
President Lincoln is lauded for his integrity, particularly regarding his management of the Civil War. All his decisions sprang from his belief that maintaining the American form of government, as expressed through the Union, was vital. His refusal to deal with dishonest people was just an extension of that natural integrity.