Controllers in the tower observed the airplane porpoise on each of the two prior touch-and-go landings. The pilot reported that on the third touch-and-go he realized he was too high and fast on final, but continued the approach anyway. The landing was hard and he then performed a go-around. He told the air traffic controller that he thought his nose wheel might be flat. The pilot made a full-stop landing and taxied to the flight school, where a damaged firewall and tailskid were found.