Let f:Z₊→R be an increasing function. We say that an infinite word w is abelian f(n)-saturated if each factor of length n contains Θ(f(n)) abelian nonequivalent factors. We show that binary infinite words cannot be abelian n²-saturated, but, for any ε>0, they can be abelian n^2−ε-saturated. There is also a sequence of finite words (w_n), with |w_n|=n, such that each w_n contains at least Cn² abelian nonequivalent factors for some constant C>0. We also consider saturated words and their connection to palindromic richness in the case of equality and k-abelian equivalence.