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, 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 with 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 Balmoral bonnet 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.

Fourth Doctor
Just like his previous selves, the Fourth Doctor's initial clothing instincts were to reach for a hat after his regeneration. However, his outfit was supplied entirely from the TARDIS' wardrobe. He tried on a number of different outfits, most of them with hats. Amongst the rejected hats were a Viking helmet and a tall clown's hat. Eventually he settled on a fairly wide-brimmed fedora. Unlike the two incarnations before him, however, he fairly consistently used this fedora, or ones very like it, throughout his life.

On occasion, he substituted another hat, such as when he, too, donned a Balmoral bonnet whilst in Scotland, (DW: Terror of the Zygons) or when he wore a sombrero when he thought he was arriving in Benidorm, Spain. (DWM: Doctor Who and the Star Beast)

Fifth Doctor
Like the incarnation before him, the Fifth Doctor had a regular hat, which he wore throughout his life. From his first adventure, he carried with him a roll-up Panama hat, with a hat band of the same colours as the rest of his outfit. He tended to wear it upon first exiting the TARDIS in a new location, but often quickly rolled it up and put it back in his jacket.

Seventh Doctor
Like the Fifth Doctor, the Seventh Doctor also had a Panama hat, though it was of a design different from that worn by his fifth self. Of all the Doctor's incarnations, however, the Seventh tended to wear his hat on almost all occasions. He was very rarely seen without it.

Eighth Doctor
The Eighth Doctor did not typically wear a hat, but he did very rarely dabble with one, if he felt it was period- and/or place-appropriate. His choice of headwear presaged that eventually worn with greater regularity by the Eleventh Doctor. Once, he emerged from the TARDIS wearing a fez when he believed he had materialised in Egypt. (DWM: Doctor Who and the Nightmare Game) On another occasion, when visiting the Lakota Sioux Nation, he wore a stetson given him by Samuel Clemens. (DWM: Bad Blood)

Eleventh Doctor
The Eleventh Doctor wore a variety of hats. While in the National Museum, he found a fez of which he became instantly fond. The fez was later removed by Amy Pond and destroyed by River Song. Undeterred, he noted to himself that it would be rather easy to buy another one. (DW: The Big Bang, The Impossible Astronaut)

Two hundred years later, Craig Owens gave him a stetson for an upcoming trip to the United States. (DW: Closing Time) He was last seen wearing it inside the Teselecta. (DW: The Wedding of River Song).

This version of the Doctor also owned a top hat, which he wore at Amy and Rory's wedding, (DW: The Big Bang) after having been poisoned by River in Nazi Germany (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).

The companions' hats
Those companions attached to UNIT or other armed forces wore hats as a part of their uniforms. Ben Jackson's sailor's cap, for instance, identified him as a member of the crew of the HMS Teazer. UNIT officers, like the Brigadier and Captain Mike Yates, most often wore a rigid, green-and-red officer's cap, similar to the British Army. Lieutenant Harry Sullivan's black, white and gold hat — otherwise quite similar to the standard Army issue — betrayed his secondment from the British Navy. Sergeant John Benton most often wore a beret in the field. Indeed, UNIT soldiers and officers alike wore berets in the field. Over the years, the standard-issue UNIT field beret changed from chocolate brown (DW: The Ambassadors of Death) to khaki green (DW: Terror of the Autons) to United Nations blue (DW: Battlefield) to red. (DW: Aliens of London, Planet of the Dead)

Although not in the military, a couple of companions nevertheless had headwear that was a part of a uniform. Tegan Jovanka had a triangular cloth cap as a part of her stewardess' uniform, but she largely abandoned its use after meeting the Fourth Doctor and Adric. (DW: Logopolis) Likewise, while Nyssa of Traken was "in uniform" as a Trakenite royal, she wore a tiara. (DW: The Keeper of Traken)

These exceptions aside, most companions simply didn't wear hats on anything like a regular basis. A few assistants did, however, wear the occasional hat strictly as a fashion accessory, often whimsical.

In both her incarnations, Romana was fond of wearing hats. Romana I wore a flat cap on her first visit to Earth (DW: The Stones of Blood) and an elaborate purple-and-green hat that was "what everyone . . . [was] wearing" on Tara. (DW: The Androids of Tara) Romana II briefly toyed with wearing the Fourth Doctor's fedora after her regeneration. She later wore a boater when punting in Cambridge (DW: Shada) and a straw sun hat whilst on a beach. (DW: The Leisure Hive)

Liz Shaw generally didn't go in for hats, but did once wear a white, broad-brimmed hat to drive Bessie. (DW: The Ambassadors of Death) Likewise, Sarah Jane Smith was mostly hat-less but did wear a casual spring hat whilst investigating Think Tank. (DW: Robot)