Hat

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

Fashion
Stella Bopbates was famous for designing hats around the year 200,100. Rose Tyler was asked this on the Game Station's version of The Weakest Link. (TV: Bad Wolf)

Worn by the Doctor
In many of their lifetimes, the Doctor wore hats, either as occasional style or as regular attire.

The First Doctor was wearing his favoured hat, an Astrakhan, when he met his first human companions, Ian and Barbara. (TV: An Unearthly Child) The Second Doctor had an early tendency to don and remark upon any hat he came across in the course of an adventure. (TV: The Power of the Daleks, The Highlanders, The Underwater Menace)

Later, the Fourth Doctor had a particular hat, a fedora, which he wore quite regularly, (TV: Robot et al.) and the Fifth (TV: Castrovalva) and Seventh Doctors (TV: Time and the Rani et al.) both favoured panama hats, each to their own style.

On noticing the Fifth Doctor actually kept a hat on his hat stand in the TARDIS, the Tenth Doctor noted that he couldn't remember the last time he actually wore a hat. (AUDIO: The Gates of Hell)

The Eleventh Doctor would later find renewed enthusiasm in hats, which he would regularly pick up and don in an adventure, (TV: The Big Bang, The Impossible Astronaut, Let's Kill Hitler) like the Second Doctor. (TV: The Power of the Daleks et al.) The Thirteenth Doctor also expressed fondness for fezzes, (TV: Kerblam!) a particular favourite for the Eleventh Doctor. (TV: The Big Bang, A Christmas Carol, The Day of the Doctor)

Worn by companions
Some of the Doctor's companions also wore hats on a regular or occasional basis.

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. Surgeon-Lieutenant Harry Sullivan's black, white and gold hat — otherwise quite similar to the standard Army issue — betrayed his secondment from the Royal 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 (TV: The Ambassadors of Death) to khaki green (TV: Terror of the Autons) to United Nations blue (TV: Battlefield) to red. (TV: 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. (TV: Logopolis) Likewise, while Nyssa of Traken was "in uniform" as a Trakenite royal, she wore a tiara. (TV: The Keeper of Traken)

These exceptions aside, most companions simply didn't wear hats on anything like a regular basis. A few wore 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 (TV: The Stones of Blood) and an elaborate purple-and-green hat that was "what everyone . . . [was] wearing" on Tara. (TV: 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 (TV: The Five Doctors) and a straw sun hat whilst on a beach. (TV: The Leisure Hive)

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

Others
Tom Cardwell wore a tin foil hat, which according to Lucie Miller, was "not a good sign at all". (AUDIO: Blood of the Daleks)

Graham O'Brien described the Cyber-helmet as a hat. (TV: The Timeless Children)

Other references
Claiming that she could save the universe from the final Flux event, the Thirteenth Doctor told the Division's Ood that she was "very good at pulling rabbits out of hats." When the Ood stated that he had neither rabbits or hats, the Doctor explained that it was a metaphor. (TV: Survivors of the Flux)