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Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity mathematically described by fractal geometry. In recent years, the engineering of self-similar structures in photonics and nano-optics technology enabled the manipulation of light states beyond periodic or disordered systems, adding novel functionalities to complex optical media with applications to nano-devices and metamaterials. Here, we extend the reach of fractal "photonics" by experimentally demonstrating multifractality of light in engineered arrays of dielectric nanoparticles. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure-property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects that emerge directly from fundamental structures of algebraic number theory.