We study the problem of exploration in stochastic Multi-Armed Bandits. Even in the simplest setting of identifying the best arm, there remains a logarithmic multiplicative gap between the known lower and upper bounds for the number of arm pulls required for the task. This extra logarithmic factor is quite meaningful in nowadays large-scale applications. We present two novel, parameter-free algorithms for identifying the best arm, in two different settings: given a target confidence and given a target budget of arm pulls, for which we prove upper bounds whose gap from the lower bound is only doubly-logarithmic in the problem parameters. We corroborate our theoretical results with experiments demonstrating that our algorithm outperforms the state-of-the-art and scales better as the size of the problem increases.