Not only is it a one-joke characterization, the joke is on the level of a below-average knock-knock joke.
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While it is one or two steps behind "Dr. Strangelove," "The President's Analyst" (1967) is a very good black comedy sniggering at Cold War paranoid. Maybe it's not as ruthless as that great comedy, but the movie romps cheerfully on its subjects with a take-no-prisoner attitude. And during this loony joy ride we eventually discover that the movie foretold something very accurate more than 40 years ago.
We have seen many psychiatrist whose lives become more burdensome than usual thanks to their unusual patients in the movies ("Analyze This") and TV series("The Sopranos" and "In Treatment"), but I think no one can top our hero Dr. Sidney Schaefer (James Coburn). His new patient is none other than the president of the United States, the most powerful figure in the world who incidentally does not appear on the screen.