The pilot and his passenger prepared for a night flight in the flying club’s airplane to maintain the pilot’s currency. He tried to start the engine three times, priming the engine a total of 11 times, and “pumping” the throttle during one attempt, but it did not start. Concerned that he may have flooded the engine, he waited 5 minutes before attempting another start with full throttle and mixture at idle/cutoff. After this attempt, the passenger noted smoke and he realized that the engine was on fire. He told his passenger to egress and attempted to extinguish the fire with the onboard fire extinguisher; however, the fire substantially damaged the airplane’s the fuselage, engine mounts, and firewall. The pilot reported no mechanical malfunctions or failures with the airplane that would have precluded normal operation. The Pilot’s Operating Handbook for the airplane suggested priming 1 to 3 times before engine start.