A large literature in psychology, and more recently in economics, has argued that monetary rewards can reduce intrinsic motivation. We investigate whether the negative impact persists when intrinsic motivation is strong, and test this hypothesis experimentally focusing on the motivation to undertake interesting and challenging tasks, informative about individual ability. We find that this type of task can generate strong intrinsic motivation, that is impervious to the effect of monetary incentives, particularly when the individual is "racing against himself". In our experiments, monetary incentives have no significant impact on performance. In a second experiment using the same kind of task but a setting designed to weaken intrinsic motivation, monetary incentives do have a significant, positive, effect on performance. This result confirms that our experimental setup may, with appropriate conditions, replicate the known crowding out effects.