Silkies. I kept a group of Silkies, just to brood for me. With 10 hens, I always had at least one broody year-round and last spring, I had my record of 8 being broody all at once! My hens would lay between 10-12 eggs, go broody, raise out their chicks. About 4-weeks after the chicks hatched, she'd start laying again, lay her 10-12 eggs over a 3 week period, go broody again, raise out chicks, repeat all year long.
I've heard bantam Cochins are also very broody; I just don't have any personal experience with them. A lot of bantam breeds are frequent broodies: Old English Game bantams, Japanese, etc.