I made a tiny woolly bobble hat! It is about 19mm in diameter (60mm circumference) and 15mm tall (not including bobble). I used double crochet stitch (that's single crochet if you're American) for most of it and treble crochet (double crochet if you are American) to give it a ridged edging. It is 38 stitches around at the widest point. The tip of my (slightly scarred) index finger is in the photos for scale.
Here it is actual size (assuming you are viewing this the same size as I am).
I've worked out that if it is supposed to be an adult sized hat then it is between 1:10 and 1:9 scale. To be honest I think this is as small as I can go - once the stitches get smaller than 1mm square, my eyes can't cope anymore!