Core Entropy is a function aimed at measuring the complexity of the iterations of post critically finite polynomials. In this talk we first recall the basic notions of polynomial dynamics: Julia sets, Hubbard Trees, and Core Entropy, and then we explore the core entropy as a function on the parameter space of polynomials. We then introduce new techniques to understand the maxima of core entropy. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).