Astrakhan

An Astrakhan was a wool hat with two points on the top. The First Doctor wore one. (TV: An Unearthly Child, PROSE: A Big Hand for the Doctor, The Massacre) In one account it was called an "oddly shaped fur hat". (PROSE: Doctor Who and an Unearthly Child)

When the Eighth Doctor saw his possible future incarnations in a Tomorrow Window, one of them was a kindly-faced old gentleman in an astrakhan hat. (PROSE: The Tomorrow Windows) The Eleventh Doctor found a different one during his adventure to collect and assemble the pieces of the Eternity Clock scattered throughout time, commenting he must have always been in a bad mood in his original incarnation because of how itchy it was. (GAME: The Eternity Clock)

Behind the scenes

 * The type of hat the First Doctor wore was never mentioned on-screen, only in the prose stories.
 * According to The Brilliant Book 2012, the Eleventh Doctor considered wearing one, even having the thought of going back in time to steal it from himself, something which River Song disapproved of.