Reward-free exploration is a reinforcement learning setting recently studied by Jin et al., who address it by running several algorithms with regret guarantees in parallel. In our work, we instead propose a more adaptive approach for reward-free exploration which directly reduces upper bounds on the maximum MDP estimation error. We show that, interestingly, our reward-free UCRL algorithm can be seen as a variant of an algorithm of Fiechter from 1994, originally proposed for a different objective that we call best-policy identification. We prove that RF-UCRL needs $\mathcal{O}(\frac{SAH^4}{\varepsilon^2}\log(\frac{1}{\delta}))$ episodes to output, with probability $1−\delta$, an $\varepsilon$-approximation of the optimal policy for any reward function. We empirically compare it to oracle strategies using a generative model.