My search for pentagonal hats also turned up this item, which isn’t exactly a hat, but Ms. Van Den is wearing it on her head, and I’ll take what I can get.
Jess Van Den of Epheriell Designs was crocheting a granny square, which normally has a D4 symmetry, as its name implies. It begins with a center that has four groups of connected stitches. But she slipped up and began with five groups instead.
Addendum: Ms. Van Den did eventually turn the pentagonal object into a hat. Picture here.