This Alice & Jasper photo contains
boater, leghorn, panama, panama hat, sailor, skimmer, straw hat, fedora, felt hat, homburg, stetson, trilby, dress hat, high hat, opera hat, silk hat, stovepipe, top hat, topper, and beaver.
There might also be
pith hat, pith helmet, sun helmet, topee, topi, campaign hat, cavalier hat, and slouch hat.