“It is the first person and the last person the president talks to, outside the vice president,” said Rahm Emanuel, Mr. Obama’s first and at 20 months his longest-serving chief of staff, and now the mayor of Chicago. “And it is both a chief — meaning top dog — but it is also staff.”

“I would argue that they are the second-most-powerful person really in the free world, behind the president and occasionally a vice president like Dick Cheney,” said David B. Cohen, a political scientist at the University of Akron who is writing a book on White House chiefs of staff. “It is the most powerful unelected position. Chiefs of staff are responsible for everything — I mean everything — that comes into the White House.”

Image Ronald A. Klain was chief of staff to two vice presidents. Credit... Beth Keiser/Associated Press

The fact that Mr. Obama has burned through four chiefs in four years — a contrast with Treasury, where Mr. Geithner stayed for the entire term and does not leave until Jan. 25, after the president’s second-term inaugural — is a testament to the unique difficulties and pressures of a job that really became institutionalized only with the Nixon administration. The chief of staff has cabinet status in this administration but ultimately is defined both by the times and the personalities involved.

While the turnover has been greater than for past presidents, Mr. Obama’s term was complicated by a global economic crisis and two wars. The musical chairs also seem to reflect Mr. Obama’s own shifting conception of what he wants from the job, however.