Economic and decision-making theories suppose that people would disengage from an extremely difficult task, because such a task does not implicate any normative utility values (i.e. success probability is almost zero). However, humans are often motivated for an extremely challenging task with little chance of success, even without any extrinsic incentives. The current study aimed to address how the nature of the task (luck vs. skill) and the presence of extrinsic rewards modulate this challenge-based motivation, and its neural correlates. Participants played a game-like, skill-based task with three different probabilities of success (i.e., high, moderate, and extremely-low chance of success), in one group without performance-based rewards (no-reward group) and the other group with performance-based monetary rewards (reward group). Participants in the third group played a similar task but the reward outcome was determined in a probabilistic manner (gambling group). Participants in the no-reward group showed increased intrinsic motivation, as the chance of success decreased (i.e. the task becomes more difficult) even if the task was almost impossible to achieve. On the other hand, the reward group exhibited the highest intrinsic motivation when the task had a moderate chance of success, and participants in the gambling group showed decreased intrinsic motivation as the chance of success decreased. The ventral striatum and the ventral pallidum also showed similar patterns of activation. These results suggest that, both at the behavioral and neural levels, people are intrinsically motivated to challenge a nearly impossible task, but only when the task requires certain skills and extrinsic rewards are not available.