That's an interesting fez you got there, Bob. The letters on the box stand for Ancient Arabic Order of the Nobles of the Mystic Shrine (AAONMS) or better known as the Shriners. It is their logo on the fez as well of course. Excellent that you've found one with the box. Note that it wasn't sold by a hatter but by a retailer of "costumes and paraphernalia".^^^^^ That is much cooler that the one that just arrived at my door. It does have the box, the the interior has no sweat band or size marked. It is too small for my size 7-1/2 noggin.
The seller (Horstmann) was established in 1815 and was under this name from 1893 to 1940. So it's at least 82 years old -- and the graphics on the box appear older to my eye, but what do I know?
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