This is not the sort of project that would normally appeal to me, but it was a special request by a friend who positively assured me that she would actually wear such a hat. More than once. In public.
In a what-was-I-thinking-in-the-middle-of-the-night moment, I started knitting with worsted weight yarn on US 2 needles, producing a fabric which stands on its own and could possibly be substituted for Kevlar. To make things worse, I used dpn's because I didn't have any circular needles in that size. Very soon I had to move one quarter of the stitches to a holder because they were falling off the needles, and I became rather adept at substituting needle for holder on every round.
In the end I'm pleased with the hat, and it actually fits! However, I learned a couple of valuable lessons: (1) use appropriate tools for the job, and (2) when something is going badly right from the start, rip it out and start again.