The quintessential "princess hat" of the Middle Ages, the hennin, was inspired by the willow-withe and felt Boqta (Ku-Ku) of Mongolian Queens, according to a blog entry for Medieval PoC. The Mongolian hats could reach up to 7 feet in height and may have served to distinguish men from women at a distance.
The article quotes author Jack Weatherford:
The contraption struck many foreign visitors as odd, but the Mongol Empire had enjoyed such prestige that medieval women of Europe imitated it with the hennin, a large cone-shaped headdress that sat towards the back of the head rather than rising straight up from it as among the Mongols. With no good source of peacock feathers, European noblewomen generally substituted gauzy streamers flowing in the wind at the top.