The father of all Turkmen* (bashi literally means "head of"), Saparmurad Niyazov (he seems to have shunned that all-too-ordinary family name lately), the first and supposedly lifelong President of Turkmenistan. His building-size portraits decorate the streets, and his gold-coated 50m-tall statue, rotating after the sun, stands on the square of his name in Ashgabat. Also, the Turkmen have adopted the great old tradition of putting presidents on the bank notes, but they have only one to choose from.
Turkmenbashi is also the new name of a port at the Caspian Sea, as well as the new name of January (yes, the month) in Turkmenistan, the name of a few streets and a highway, the name of an oil-processing plant, well, you guessed the rest. His book Rukhnama is studied in schools and generally everywhere as the source of all wisdom there is for Turkmen.
The http://turkmenbashi.org website, strangely, represents the Embassy of Turkmenistan in the U.S. It seems that Turkmenbashi is the state, and even the sun dares to rise only with his gracious permit (he's been heard to claim he controls the weather over his country, too).
Some not-so-marvellous bits of trivia: had troubled childhood, worked in Leningrad on an industrial plant, headed the Communist Party of Turkmenistan before the collapse of USSR.
*Do not confuse with Turks, the natives of Turkey.
Thanks Gritchka and JudyT for hints and general interest in the subject.