The Taub-NUT metric is a much more classical prototype. Its asymptotic geometry is an $S^{1}$-bundle over $\mathbb{R}^{3}$, with non-maximal volume growth $Vol(B(r))=O(r^{3})$. Algebro-geometrically the Taub-NUT is associated with the fibration $\mathbb{C}^{2}\xrightarrow{z_{1}z_{2}}\mathbb{C}$, whose fibres are cylinders. The holomorphic function $z_{1}z_{2}$ has polynomial growth, while $z_{1},z_{2}$ individually have exponential growth.