The Librarian of Congress
The Library of Congress is a source of several constitutional curiosities. One is that the Librarian of Congress is nominated by the President, confirmed by the Senate, and serves for life. As far as I can tell, this makes him (only men have been the Librarian of Congress) the only non-Article III constitutional officer with life tenure. The Library has been around since 1800, but there have been only 13 Librarians. (There have been 17 Chief Justices of the Supreme Court, for all of you keeping score at home.)
Another issue is the source of Congress’s authority to create the Library in the first place. The answer seems to be that creating what was essentially a research resource for Congress was considered part of its inherent power; there was no enumerated power involved (unless, I suppose, you say that because the Library was situated in the District of Columbia that fell within Congress’s plenary power there).
UPDATE: Marty Lederman, who is THE expert on all things Executive Branch, points out in the comments that the Librarian actually can be removed by the President–it just hasn’t happened since Andrew Jackson’s day.