Stovepipe hat

A stovepipe or stove-pipe hat was an unusually tall top hat favoured by the newly-regenerated second Doctor. He apparently became disenchanted with the hat, though, as he rarely, if ever, wore it following his adventures with Ben, Polly and Jamie in Atlantis. (DW: The Underwater Menace)
 * Though the televised series never calls it a "stovepipe hat", at least a couple of works of prose fiction — PDA: The Murder Game and TN: Wonderland — do. Nevertheless, some fashion experts might disagree that it's actually a stovepipe, as it doesn't have a flat top.