An executive order isn't that complicated: it's basically just the president telling people to do something.
Executive orders are simply presidential directives issued to agents of the executive department by its boss.
Brian R. Dirck (2007). The Executive Branch of Federal Government: People, Process, and Politics, p.102
So could the President issue an executive order which declared himself President for Life? Sure – that would be called a coup, and what happened next would depend on whether people chose to follow that order and support the coup, or reject it.
Technically, the Supreme Court has held that Executive Orders must be rooted either in powers granted by Article II of the Constitution or statutes passed by Congress, but ultimately that decision comes down to individual people who will have to decide whether to follow such an order or disobey.