Most men will assume a woman thinks they're suffering from male pattern baldness if she asks for a photo where he is not wearing a hat.
It'd be laughable to think that thought isn't in her mind. Of course it is. Every time. May not be the ONLY thing, but it certainly would be.
What I have noticed when wearing a hat is how much of a difference people see me when I take it off......I think it is a surprise to them, a disappointment, a shock......
Depends what type of hat. If it's covering up more than just their hair, and covering their face -- yeah. It can be a "Why are you wearing that hat? I can see you now." But pics with a gal in a baseball cap -- I can see what she looks like just fine. Are guys going "Hey babe, you only have a couple photos. They're clear, they're fine -- but you're wearing a baseball cap, even though it's not at all covering up your face one bit. Can you give me one without it? I need to know."? No. Why would a gal ask the same thing? Hair line. :)