Fez

A fez was a type of brimless hat in the shape of a flat-topped cone. Fezzes were made of felt and typically, but not always, had a single tassel attached to the center of their flat tops. It was particularly popular in Egypt, judging from the number of times the Doctor encountered fezzes around Egyptians or Egyptian artifacts.

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

The Seventh Doctor briefly donned a fez while searching through the basement of Windsor Castle, before passing it on to Ace (and also picked up a mop, for that matter). (DW: Silver Nemesis)

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 lost the fez soon after the police began to chase him from the stadium. (DWM: Doctor Who and the Nightmare Game)

Whilst in his eleventh incarnation, 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 by asking, "What in the name of sanity have you got on your head?" he answered, "It's a fez. I wear a fez now. Fezzes are cool.", much as he did when defending his bow tie. River Song 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)

When the Doctor dropped his fez over the TARDIS' controls, it resulted in the fez' original owner, Albert Einstein, materialising via a Time window. (DW: Death Is the Only Answer)

While taking Abigail Pettigrew out one Christmas Eve, the Eleventh Doctor and Kazran Sardick donned fezzes. (DW: A Christmas Carol)

The Doctor considered sporting a fez when he, Amy, and Rory visited Nevada in 1861. However, Amy and Rory insisted against it, so the Doctor wore a stetson instead. (NSA: The Good, the Bad and the Alien)

When a version of Rory from the future appeared, the Doctor used a Fez taken from the TARDIS storeroom to differentiate him from the present Rory. (NSA: Touched by an Angel)

The Doctor claimed he looked good in a fez. (NSA: The Way Through the Woods)

The Eleventh Doctor wore a fez while dancing in a Laurel &amp; Hardy film. He later requested a fez while in the White House. His request was ignored. Later, he blamed his inability to do the work assigned to him as well as he had promised on his lack of a fez. (DW: The Impossible Astronaut)

Rory later mentioned to an older version of Amy that the glasses he had to wear were better than a fez. (DW: The Girl Who Waited)

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 would evolve 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 Series 5 episode guides in The Brilliant Book 2011 feature items which relate to each episode. A fez is featured on the page for The Big Bang.
 * The fez worn in The Big Bang appears to be lacking a tassel, which is often associated with the hat.