Let \(\Theta \subseteq [0,1]\) be an interval, let \(Y_0,Y_1\) be Banach spaces and let \(\{ Y_\eta \}_{\eta \in \Theta}\) be an ordered family of Banach spaces, i.e., \(Y_0 \hookrightarrow Y_\theta \hookrightarrow Y_\eta \hookrightarrow Y_1\) for any \(\theta \leq \eta\) with \(\theta, \eta \in \Theta\). The authors define the extrapolation spaces \(Y_\theta (\log Y)_{b,q}^+\) and \(Y_\theta (\log Y)_{b,q}^-\), respectively (for the exact definitions we refer to their paper) and derive asymptotic upper estimates for the entropy numbers of the operators \(T: Y_\theta (\log Y)_{b,q}^- \rightarrow X\) and \(T: X \rightarrow Y_\theta (\log Y)_{b,q}^+\), respectively, where \(X\) is any fixed Banach space. They apply these upper estimates to describe the exact asymptotic behaviour of entropy numbers of embeddings from limiting fractional Sobolev spaces into generalized Lorentz-Zygmund spaces. Indeed, for \(\Omega\) being a bounded \(\mathcal{C}^\infty\) domain or a bounded Lipschitz domain in \(\mathbb{R}^d\), \(1 0\), it is shown that \(e_k(\mathrm{id}: H_p^{d/p}(\Omega) \rightarrow L_\infty(\log L)_{-1/{p'};-\alpha}(\Omega)) \asymp (\log k)^{-\alpha}\). The authors also prove upper estimates for entropy numbers of embeddings involving logarithmic Sobolev spaces and Besov spaces, respectively, which in some instances turn out to be optimal up to a logarithmic factor.