The reinforcement learning community has recently intensified its interest in online planning methods, due to their relative independence on the state space size. However, tight near-optimality guarantees are not yet available for the general case of stochastic Markov decision processes and closed-loop, state-dependent planning policies. We therefore consider an algorithm related to AO* that optimistically explores a tree representation of the space of closed-loop policies, and we analyze the near-optimality of the action it returns after n tree node expansions. While this optimistic planning requires a finite number of actions and possible next states for each transition, its asymptotic performance does not depend directly on these numbers, but only on the subset of nodes that significantly impact near-optimal policies. We characterize this set by introducing a novel measure of problem complexity, called the near-optimality exponent. Specializing the exponent and performance bound for some interesting classes of MDPs illustrates the algorithm works better when there are fewer near-optimal policies and less uniform transition probabilities.