Fez

A Fez was a type of hat with the shape of a truncated cone often found in Africa, India, and many Middle Eastern countries, such as Morroco, and was also closely associated with the "Shriners" fraternal organization.

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

The Doctor briefly donned a Fez while searching through the basement of Windsor Castle, before passing it on to Ace. (DW: Silver Nemesis)

The Doctor wore a Fez to a football game in 1977. He actually intended to land the TARDIS near the River Nile which was 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)

The Doctor took a fez off an exhibit in the museum where the Pandorica was, and put it on, wearing it as he leapt back and forth through time. When he rescued River Song out of a burning TARDIS, she asked him "What in the name of sanity have you got on your head?" and he replied "Its a fez. I wear a fez now. Fezes are cool." in the same way he often states that "bowties are cool". It was snatched off his head and hurled in the air by Amy Pond and then destroyed with a gun by River Song. When the Doctor awoke after using the Pandorica to undo the destruction of the Universe, he felt his head and was disappointed to not be wearing a fez anymore, 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.