Astrakhan

An Astrakhan was a wool hat with the two points on the top. The First Doctor wore one. (TV: An Unearthly Child) 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
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.