Hat

A hat was an article of clothing worn on the top of the head.

The Doctor's hats
Throughout the majority of his lifetimes, the Doctor wore a hat on a regular or semi-regular basis. Only a minority of incarnations — the Sixth, Eighth, Ninth and Tenth Doctors — never showed a particular interest in hats.

First Doctor
The First Doctor's usual hat of choice was a karakul, a triangular hat often associated with Afghanistan, or what one account called an "oddly shaped fur hat". (DWN: Doctor Who and an Unearthly Child) It was this hat he was wearing when he met Barbara Wright and Ian Chesterton, as well as during his first adventure with Ben Jackson and Polly Wright. (DW: An Unearthly Child, The War Machines) However, he was also known to possess a kind of Panama hat, which he wore in hotter climes. For instance, he was seen to don an apparently white Panama when he was on the hunt for the Daleks' taranium in Egypt. (DW: "Escape Switch")

This incarnation was not averse to occasionally playing dress-up. When in Revolutionary France, he obtained period-appropriate wardrobe. Because the costume was that of a regional official, his hat was particularly elaborate, bearing a strong resemblance to an early 18th century French army officer's hat. (DW: "A Change of Identity")

Second Doctor
Early on, the Second Doctor was particularly obsessed by hats. One of his very first acts after regeneration was to rummage through a trunk in the TARDIS to find a stovepipe hat to wear. He wore the hat throughout his first adventure on Vulcan. (DW: The Power of the Daleks)

He sometimes uttered the phrase, "I would like a hat like that (or this)" if he saw a hat he particularly fancied. (DW: The Power of the Daleks, The Highlanders) In fact, when he arrived in Scotland in the 18th century, he happened across a Tam o' Shanter on a field of battle, and immediately donned it. Later in the same adventure, he wore a British Army tricorn as a disguise.

On a trip to Atlantis, he continued to show his penchant for disguise, and temporarily wore a gypsy headdress to appear as a merchant. (DW: The Underwater Menace)

However, he seemed to "grow out of" hats as he matured, mostly abandoning hats altogether after Atlantis. Some accounts, however, insisted that he continued wearing the stovepipe hat on a very regular basis, even up to the time that he travelled with Zoe Heriot. (TVC: Most stories; DWAN: Most stories)
 * It's unclear why World Distributors and Polystyle continued to portray him with a stovepipe, when he actually only wore a stovepipe in The Power of the Daleks.

Third Doctor
Like his predecessor, the Third Doctor almost immediately donned a hat after his regeneration cycle ended. He found a trilby amongst a physician's clothes at the Ashbridge Cottage Hospital. When he tried it on, and looked in a mirror, he seemed to give it a look of approval. (DW: Spearhead from Space) However, he only briefly wore it and generally was without a hat for the rest of his life.

Eleventh Doctor
The Eleventh Doctor was fond of a variety of hats.

While in the National Museum, the Eleventh Doctor found a fez which he became very fond of. The fez was later removed by Amy Pond and destroyed by River Song. He sought another fez after this. (DW: The Big Bang, DW: The Impossible Astronaut)

The Eleventh Doctor acquired a Stetson hat from Craig Owens.(DW: Closing Time) He was last seen wearing it inside the Teselecta. (DW: The Wedding of River Song).

The Eleventh Doctor owned a top hat, which he wore at Amy and Rory's wedding, after having been poisoned by River (DW: Let's Kill Hitler) and again to conceal a device that allowed him to lure the Squall back into their dimension of origin (NSA: Paradox Lost).