Hand armor
Maybe this is ridiculously obvious, but having taken a couple of tumbles lately, it seems worth mentioning.
Biking gloves do a lot more than just absorb shock from the handlebars.
They also protect your hands if you fall. Think about it. When you fall, what almost always hits the asphalt first and takes most of the blow. Yup- your hands.
About a month ago, I was in a bike crash (involving a car). I wasn't wearing my gloves because my handlebars had good padding. I tore up my hand and it took two weeks for the skin to grow back.
A couple of nights ago, I was wearing my gloves - having learned the hard way, and crashed. The parts of my fingers with no gloves got really chewed up, but the pads of my hands are OK.
So, there you have it. I recommend wearing gloves as a compliment to the helmet you use to protect yourself.