Would the man with without a hat had a hat, he would be the man without with hat. Alternatively you can call him the man without without hat. Then "without without hat" runs principially on the same out as the double vernonung. If the man with without a hat had really no hat, then you could take as synonym for "with without hat" also "with with hat"; we remember at the third sentence of this post. If the man without hat had a hat, then he is the man without with hat. If the man with hat had no hat, then we are at the beginning of the lyre: The man with without hat.
Bei dem Weg, it gives three theories about "with without hat". Look at my Gewinde "Der Mann mit ohne Hut".