I started with some Cascade 220, color 4008. Knowing that I am a relaxed (very relaxed) knitter, I used 2.75mm (US2) needles for the ribbing and 3.75mm (US5) needles for the body, sizes I usually use with worsted-weight yarn. The result is a nice hat, but not a beret.
Could I make it more floppy and beret-like? Yes, with bigger needles. I retained the 2.75mm needles for the ribbing but went to 4.5mm (US7) for the body. The yarn is Plymouth Galway Colornet, color 565. Ta-da! A beret!
The red hat has a finished weight of 84 grams. The green one has a finished weight of 102 grams. That's a difference of 22 percent!
And, yes, I know that a hat trick needs three. So here's the third hat, based on Jared Flood's Turn a Square hat. I like the shaping on this hat. The blue is Brown Sheep Top of the Lamb in Blue Flannel; the stripes are Cascade 220 in color 8404 (brown) and color 8021 (off-white). Needles are US3 for the ribbing and US4 for the body. Yes, relaxed knitter.