The pilot reported that an approach was being made in light crosswind conditions. Touchdown was normal, but the aircraft then became airborne again. A series of bounces then followed, each larger than the last. At some point during this phase the nose strut oleo collapsed and the propeller struck the ground. The aircraft swerved off the runway and slowed to the point where the pilot was able to regain control. Subsequent examination revealed damage to the engine firewall and lower fuselage structure. The pilot was evidently surprised by the speed and severity of the bounces, and had apparently not considered making a go around. This accident was not the subject of an on scene investigation.