The pilot reported landing "flat" and the airplane "bounced." After a second "bounce" the pilot initiated a go-around. However, when the nose landing gear contacted the runway a third time, the nose wheel "blew out." The airplane veered to the left, exited the runway pavement, and nosed over. The airplane sustained substantial damage to the fuselage. The pilot observed no anomalies during the preflight inspection. A flight instructor who witnessed the accident stated that the airplane remained in a "nose down pitch attitude" until the nose wheel contacted the runway which caused the airplane to "bounce." After the third bounce, the nose wheel "blew out sending white smoke into the air."