The stereotypical chef beloved of TV, cartoons, and elsewhere is always pictured wearing a high white hat.
A hat I can understand. But why so high?
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Reaching back to my time in culinary school at Johnson & Wales University in Providence, RI:
The style of hat originated in the Greek orthodox monastaries during the middle ages where chefs (often taking refuge with other educated members of society) would distinguish themselves from the monks by wearing a white hat (monks wearing black). The early chef hats were slightly elevated but not as tall as they eventually became.
Marie Antoine Careme (commonly called "Chef of Kings and King of Chefs") is credited as the father of classical french cuisine for the development of mother sauces and standard garniture and classification of terminology used in kitchens. He's also credited with much of the organization and division of labor in kitchens still used today. To designate role and stature in the kitchen he had his cooks wear hats of various height. His being the tallest designated him as the Head or Executive Chef.
Traditionally it is said that the now traditional pleats indicate the number of ways that the chef wearing the hat can prepare an egg (of course this predates the disposable paper hats we now use). 100 pleats is considered to be the "ultimate" number to be desired. However as a chef with 25 years experience in commercial kitchens, I'd be hard pressed to name 100 ways (without including eggs as an ingredient in recipes) to prepare an egg.
In summary, the old chefs in the greek orthodox used to hide among the clergy and changed the color to white in an attempt to appease the wrath of god, and Henry the VII beheaded a chef for finding hair in soup and made the next guy wear a hat.
I would say that now it's mainly because of tradition. Most chefs I know wear their hair up in a bandana unless they are on a carving station.
Raised chef hats were naturally developed because when a chef/ or other cooking staff wearing no hat or or only a cap , stood up up after getting obtaining something low or attending something requiring being a bent over position and then and began to rise up, and something like a open cupboard door or a hanging utensil (heavy knives, skewers, cleavers) above would conk him in the head, causing injuries, and adding even more tension to the organization of meal preparation I know we have all done it and how much it hurts. So the hats we have now sort of first evolved over the centuries in a sort of Darwinian genesis of added heights to foreworn the wearer of the clean white hat,an impending hit to the noggin and worked to that purpose very well, for many centuries now. Of course the white hat also signified proper sanitary conditions were been followed and also kept long hairs under the hat from migrating into the meals being prepared.