Fez

A fez was a type of brimless hat with a flat top. Usually made of red felt, fezzes typically, but not always, had a single tassel attached to the center of their flat tops. Judging from the number of times the Doctor encountered fezzes around Egyptians or Egyptian artifacts, the fez seemed to be closely associated with Arabic culture on Earth.

Ibrahim Namin wore a fez. (DW: Pyramids of Mars)

The Doctor's seventh incarnation briefly donned a fez while searching through the basement of Windsor Castle, before passing it on to Ace. (DW: Silver Nemesis) Later, his eighth self wore a fez to a football game in 1977. He originally intended to land the TARDIS near the River Nile which was the reason why he wore the fez in the first place. The Doctor soon lost the fez after the police began to chase him from the stadium. (DWM: Doctor Who and the Nightmare Game)

Whilst in his eleventh body, he took a fez from an exhibit in the museum where the Pandorica was, and wore it as he leapt back and forth through time. When River Song queried this he answered, "Its a fez. I wear a fez now. Fezes are cool." Finding the answer insufficient, she and Amy Pond promptly destroyed it, before continuing to reboot the universe. Later, he noted that he was missing his fez, but concluded that he could buy a new one. (DW: The Big Bang)

Behind the scenes

 * Steven Moffat said that Matt Smith grew fond of the fez so they had to destroy it to stop him from insisting on wearing it next series. Matt Smith had also once said that he hoped the costume evolved to include a hat.
 * Following the airing of The Big Bang, Doctor Who Adventures starting running "top five" lists, with the #1 item being awarded the "Golden Fez".
 * The fez worn in The Big Bang appears to be lacking a tassel, which is often associated with the hat.