We investigate how star formation is spatially organized in the grand-design spiral NGC 1566 from deep Hubble Space Telescope photometry with the Legacy ExtraGalactic UV Survey. Our contour-based clustering analysis reveals 890 distinct stellar conglomerations at various levels of significance. These star-forming complexes are organized in a hierarchical fashion with the larger congregations consisting of smaller structures, which themselves fragment into even smaller and more compact stellar groupings. Their size distribution, covering a wide range in length-scales, shows a power law as expected from scale-free processes. We explain this shape with a simple 'fragmentation and enrichment' model. The hierarchical morphology of the complexes is confirmed by their mass-size relation that can be represented by a power law with a fractional exponent, analogous to that determined for fractal molecular clouds. The surface stellar density distribution of the complexes shows a lognormal shape similar to that for supersonic non-gravitating turbulent gas. Between 50 and 65 per cent of the recently formed stars, as well as about 90 per cent of the young star clusters, are found inside the stellar complexes, located along the spiral arms. We find an age difference between young stars inside the complexes and those in their direct vicinity in the arms of at least 10 Myr. This time-scale may relate to the minimum time for stellar evaporation, although we cannot exclude the in situ formation of stars. As expected, star formation preferentially occurs in spiral arms. Our findings reveal turbulent-driven hierarchical star formation along the arms of a grand-design galaxy.