Fastest decaying probabilities with infinite entropy

Question

A well known theorem of analysis is that there is no slowest rate of divergence of a series. "Completely irrelevantly," we know that there exists probabilities (e.g. $\mathtt{Const}/(n\log^2 n)\mathbf{1}(n \geq 3,\,n\in\mathbb{Z})$) with infinite entropy. An analogous question is then: Is there a fastest rate of decay of non-zero probabilities with infinite entropy? In the same spirit: Is there a slowest rate of decay of non-zero probabilities with finite entropy?