This was the badge of office for nearly two thousand years until the events of Sourcery, when it was presumed destroyed. Worn by over two hundred previous Archchancellors, the hat is imbued with magic and has taken on a life of its own.
It manipulates Rincewind and Conina to take it to Klatch, out of the reach of the Sourcerer. Once in Klatch, it finds its way to Abrim, the Grand Vizier, a man who was frustrated in his first preferred career as a wizard by having been discovered to be too mentally unstable even for Unseen University. Abrim and the Hat fight a sucessful war against Coin until a lucky shot from Marmaric Carding gets through (admittedly, while Abrim is distracted) and destroys the wizards' Tower in Al-Khali.
The Hat is missing, presumed de-millinered. Subsequent Archchancellors appear to have managed perfectly well without, and anyway Mustrum Ridcully, an Archchancellor with a different set of priorities, would spurn it because there'd be nowhere for his fishing flies to go, and no handy little bit at the top that unscrews into a cup just big enough for a bracing shot of Bentinck's Very Old Peculiar Brandy, from the bottle tailored to fit inside. Indeed, the new Archchancellor's Hat also incorporates a cunningly tailored roll of oiled silk and four telescoping legs, which when extended produce a compact and serviceable tent. The new-model hat also incorporates pockets containing three days' iron rations, and an elasticated band holding a very small pistol crossbow, together with ample room for fishing flies. It has also been capacious enough for a mouse on a treadmill and various unspeakable nostrums against the male-pattern baldness Ridcully denies he has got.. (crop-circles of the scalp)
As the Archchancellor's Hat will, by its very nature, store up the accumulated magical wisdom of all who wear it, one wonders what essence of Ridcully is being infused in the fabric for the next wearer... Note:In Unseen Academicals it is revealed Ridully DOES possess the Archchancellor's hat "worn through the ages", however he informs the Patrician that he does not like it as it "grumbles" a lot.
Shades here - although it precedes it - of the Sorting Hat from Ms Rowling's stories about a boy wizard...