The most valuable coin in the world is undoubtedly the Facing Head Pantikapaion Stater. Although it is not the most expensive coin (it almost is if you ignore the inflated prices of early American coins), it can be considered the most valuable, culturally, and artistically. The Stater was sold from the great Prospero collection in a 2011 auction for 3.25 million dollars.
Pantikapaion (or Panticapaeum) was a Greek colony in Bosporus on the Black Sea relatively far east. Grave gifts in the tombs show Pantikapaion to have been a wealthy city (often accredited to grain exports), and it is known that they imported pottery from Athens. The reverse depicts the standard symbols of a griffin holding a spear, a grain-ear, a sickle and the legend ΠΑΝ (Pan). The obverse depicts the head of a satyr with horse ears (possibly Pan himself), and accounts for the coin's value. Godfrey Locker Lampson wrote of his example: ‘The head of the satyr is a marvel of speaking portraiture. That so much expression could be packed into so small a round would not be believed by any one (sic) who had not seen it’.
The provenance of the coin is somewhat lacking, however. It was auctioned from the Prospero collection, and before that was purchased from the Swiss Bank, Ban Leu (presumably representing an anonymous collector). We are fortunate to have witnessed it before it disappears into another private collection for what could be an eternity.
Post Scriptum: Locker Lampson’s near identical example came from the Grand Duke Alexander Mikhailovich collection and now rests in the Gulbenkian museum with the collection of the wealthy oil baron Calouste Gulbenkian along with two other examples, one of which shares the same obverse die. As far as I can tell, the Gulbenkian museum has not published their examples outside of 'A Catalogue of the Calouste Gulbenkian Collection of Greek Coins', if you can find a copy.