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As has long been known to computer scientists, the performance of probabilistic algorithms characterized by relatively large runtime fluctuations can be improved by applying a restart, i.e., episodic interruption of a randomized computational procedure followed by initialization of its new statistically independent realization. A similar effect of restart-induced process acceleration could potentially be possible in the context of enzymatic reactions, where dissociation of the enzyme-substrate intermediate corresponds to restarting the catalytic step of the reaction. To date, a significant number of analytical results have been obtained in physics and computer science regarding the effect of restart on the completion time statistics in various model problems, however, the fundamental limits of restart efficiency remain unknown. Here we derive a range of universal statistical inequalities that offer constraints on the effect that restart could impose on the completion time of a generic stochastic process. The corresponding bounds are expressed via simple statistical metrics of the original process such as harmonic mean $h$, median value $m$ and mode $M$, and, thus, are remarkably practical. We test our analytical predictions with multiple numerical examples, discuss implications arising from them and important avenues of future work.