President Trump announced that he would appoint himself as chairman of the Kennedy Center Board of Trustees. 20 U.S.C. § 76h provides that "The general trustees shall be appointed by the President of the United States." As a statutory matter, Trump seems to have this authority. I am unaware of any instance in which any President has ever appointed himself to an office. (Here, I will assume that the chairman position is in fact an office, but I haven't thought the issue through fully; indeed, the precise status of the Kennedy Center is apparently an open question.)
Is such a self-appointment constitutional? First, in the abstract, government officials have used their authority to obtain new offices. Governors have appointed themselves to fill Senate vacancies. As best as I can recall, these governors also resigned from their state position to avoid dual-office holding. But such a resignation was not required by the Constitution. Perhaps a related issue occurred during the impeachment of President Andrew Johnson. At the time, Benjamin Wade, the Senate President Pro Tempore, was next in line for the presidency. Wade voted to convict Johnson, though Johnson was ultimately acquitted.
Second, Trump's self-appointment does not violate the Incompatibility Clause. That provision states that "and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office." This clause does not prohibit dual office-holding within the executive and/or judicial branches. Chief Justice Marshall concurrently served as Secretary of State in the final days of the Adams Administration. (Marshall was at fault for not delivering William Marbury's commission).
