Not long after Mitt Romney dropped out of the presidential race in early 2008, a titan of New York finance, Julian H. Robertson, flew to Utah to deliver an eye-popping offer. He asked Mr. Romney to become chief executive of his hedge fund, Tiger Management, for an annual salary of about $30 million, plus investment profits, according to two people told of the discussions. For Mr. Romney, who had spent the previous decade in public life forgoing any paychecks, the position promised to catapult him back to the pinnacle of American business and into the ranks of the stratospherically rich. Several friends and relatives urged him to accept. “Let’s put it this way,” said Mr. Robertson. “He could have made a lot of money.” But Mr. Romney was uninterested.