For generic systems exhibiting power law behaviors, and hence multiscale dependencies, we propose a simple tool to analyze multifractality and intermittency, after noticing that these concepts are directly related to the deformation of a probability density function from Gaussian at large scales to non-Gaussian at smaller scales. Our framework is based on information theory and uses Shannon entropy and Kullback-Leibler divergence. We provide an extensive application to three-dimensional fully developed turbulence, seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked. We compute our quantity on experimental Eulerian velocity measurements, as well as on synthetic processes and phenomenological models of fluid turbulence. Our approach is very general and does not require any underlying model of the system, although it can probe the relevance of such a model.