Wear no hat at all, if you wish to become stronger. It is sheltering you.
... 'shelter' is kind of the /point/ of a hat. I've had heat exhaustion before - I'd rather avoid it if I can, with a nice, shading hat. I live in a place where the windchill can drop below -30; walking around without a warm hat is kind of a no-go. And the 'Crasche' safety inserts are specifically designed to help shelter my all-too-vulnerable brain from damage, and at least slightly reduce the odds of my undergoing information-theoretical death.
The three situations I just listed don't seem to be the sort of things which I can improve my resistance against by not wearing a hat.
For years, I've taken being a classic nerd, geek, and hacker as a point of pride - eg, consciously trying to judge people by the code they produce, or whatever else they write, as opposed to judging them on their appearance, to the point that I prefer /not/ to know what my favourite authors look like. I've tried to make what strengths I can out of the resulting weaknesses, such as reducing decision fatigue by keeping a single hair-style for many years, wearing whichever t-shirt is on top of the clean shirt pile, and so on.
I'm no longer satisfied with this. I want to become stronger.
Last month, I bought a dozen button-up, collared shirts... and have noticed slight, but consistent changes in my workflow when I wear them. I want to leverage whatever other clothing-based self-improvements are within my budget.
For some years, I've worn a floppy boonie hat to shade my delicate eyes from the burning rays of the sun. (I've even been seen wearing it with a photographer's vest instead of a daypack while tromping around my hometown.) I'm thinking of trying out the 'Crasche' safety inserts mentioned in the recent Open Thread while hiking far from medical help, which would require a baseball-like cap with a sweatband. Given my proclivity for taking something that works and sticking with it for years, I might be wearing that cap for a very long time.
Thus, a multi-layered question: Which hat should I buy? Which factors should I take into account... and which shouldn't I? Are there any subreddits, forums, or other online discussion groups whose members would be willing to take this question seriously and with only a minimum of mockery?