The pilot reported that after a cross country flight an airspeed of 65 knots was maintained while on final approach. When the airplane touched-down the pilot thought the landing was hard “where I flared a bit too early and stalled...But there was a total unexpected bounce.” The pilot stated she verified the throttle was idle when the airplane touched down again and bounced a second time. The pilot continued to pull back on the yoke and the airplane touched down and bounced for a third time. After the third bounce the airplane touched down and came to a stop at the end of the runway. As the pilot was taxiing the airplane back to the parking area, she noticed that when the rudder was fully deflected to the right it made a clipping sound. When she exited the airplane, she examined the airplane and observed the propeller tips were bent. Post examination of the airplane by a Federal Aviation Administration inspector revealed that the airplane sustained substantial damage to the firewall and fuselage. The pilot reported no mechanical failures or malfunctions prior to the accident.