Doc's hat is one of the most important objects relating to Doc because he regularly needs it to take on any task he has in mind. As to why his hat seemingly hold an endless array of gadgets, Doc even replied that it "comes with extra trunk space" (in the episode "Buckets"). He also has a mechanical version of his hat dubbed the Tells-Me-Who's-Knocking-at-the-Fedora that can tell who's at the front door in advance.
It is revealed in "The Rock of Sages" that the hat was passed down to Doc from many generations as a heirloom. It originated from one of the Seven Sages that bore resemblance to Doc and was one of the locations that protected each part of the very powerful gem known as the Rock of Sages that led to the founding of Jollywood, and the hat is also known as the Conical Cap of Truth. However, in the Season 2 episode "Dr. Jingleheimer" Doc stated to the audience that his hat -- was stolen by the titular character to build a device to shrink the kingdom -- was his invention.
Doc's hat employed a pair of mechanical hands using a clipboard and a pencil to record findings for the source of Sneezy's severe allergies. Later in the episode, the hand shows the dwarfs the true source -- Bob.
The magnifying glass is also used to examine the extent of Starchy's leg injury.
The hat becomes a scale to weigh gems at the mine.
Doc inspects the Royal Ruby, then a whistle signaled the end of the work day.
The mechanical hand made another appearance using a feather to force Sneezy to blow Delightful's pipe organ out of its room to make way for a new one.
The mechanical hand is used to hurl water balloons at Grim the dragon.
Doc's hat employed a net to catch all the cats and dogs cascading from the sky, a work of the Glooms.
At first, the mechanical hand pulls out the Fiddle-dee Foodle-dee Phantom Finder, and then an elephant's trunk blows Lord Starchbottom away after Doc remarked that his hat has ample space. Later, five sandwiches on platters are handed out to rest of the 7D (except Bashful). Curious, Starchbottom begins to take a gander inside the hat, but he sees a woman taking a shower, leaving him embarrassed.
As part of preparing and grooming Roar for his upcoming recital, a brush is used to clean his teeth. Later, the pair of hands appear as a part of Doc's reaction to Roar's sudden rampage.
A telescope appeared to show Sneezy where the Big Bad Wolf is headed next.
The mechanical hand with a pencil and clipboard appeared again to jot down notes on how the thingy should be.
The mechanical hand gives out a handkerchief after Sneezy sneezed on Doc's plans.
A pair of hands appears when Doc laments he has to search for more quiet places.
The handkerchief appeared again to wipe Doc's glasses.
The lantern again appeared to help Doc see working on his plans for his thingy.
A propeller comes out from the top of his hat to fly him and Sneezy to Memory Lane.
The mechanical hands participated in a game of volleyball while Doc reads.
A light bulb came out of the hat so Doc can learn about the Bing Bong beans underground.
In "The Rock of Sages," Doc's hat is also known as the Conical Cap of Truth, the location of the indigo gem.