I went out with someone who was incredibly bright (*yeah yeah no jokes please! ). He got the top first at Oxford in his subject (and stopped revising the week before finals and spent the last week realaxing- bastard!). He had his first paper accepted by a journal when he was 13, and has published many many papers. He is recognised as a world expert in his area.
He went to a bog standard comprehensive; his best friend from there (who is still a friend) is a builder. They spent an awful lot of time outside school just hanging around. He didn't have extra lessons, he was just fascinated in one particular area and persued it, and still does. It's the (academic) area he works in now. I don't know whether he was ever bored at school, I'll ask him when I next see him. I do know he was removed from playschool because his mum thought it was too academic (they were teaching reading she thought that should wait until school).
Really really bright children don't necessarily need constant input - that's what distinguishes them, they just do it because they want to.