Self-locomotion is central to animal behaviour and survival. It is generally analysed by focusing on preferred speeds and gaits under particular biological and physical constraints. In the present paper we focus instead on the maximum speed and we study its order-of-magnitude scaling with body size, from bacteria to the largest terrestrial and aquatic organisms. Using data for about 460 species of various taxonomic groups, we find a maximum relative speed of the order of magnitude of ten body lengths per second over a 10 20-fold mass range of running and swimming animals. This result implies a locomotor time scale of the order of one tenth of second, virtually independent on body size, anatomy and locomotion style, whose ubiquity requires an explanation building on basic properties of motile organisms. From first-principle estimates, we relate this generic time scale to other basic biological properties, using in particular the recent generalisation of the muscle specific tension to molecular motors. Finally, we go a step further by relating this time scale to still more basic quantities, as environmental conditions at Earth in addition to fundamental physical and chemical constants.