Composition of two meromorphic functions which itself is not meromorphic?

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What are examples of two meromorphic functions in which their composition is not meromorphic? I read in another post that for the composition of two meromorphic functions f and g respectively to be meromorphic, either f must be rational or g must be holomorphic in C.

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Let $g: \mathbb{C}^{\times} \to \mathbb{C}, z \mapsto \frac{1}{z}$. Take $\exp \circ \ g$. It is not meromorphic on $\mathbb{C}$.