Broodiness is hormonal, so leaving eggs out won't necessarily induce broodiness. I have 10 Silkie pullets and someone is ALWAYS broody. Silkies are the Queens of Broodiness. Once they start laying eggs, they're like ticking time bombs, it's only a matter of time before they go broody. When they were younger, as soon as a new one starting laying, I would think "uh, oh - I'll have another broody soon!"
As for how many times a year - I haven't had them long enough to tell... The first one went broody at 5 months old and sat for 6 weeks (I tried to wait her out - she won), hatched her chicks and didn't start laying again for another 6 weeks or so. She's only been laying again for about 6 weeks - so I'd imagine she'll be broody again soon!
I also have 6 EEs almost a year old now and none of them have shown any broody tendencies. A friend of mine has EEs, Orpingtons and Australorps for the last 3 years and never had a broody.