An entire function that violates quasiminimality

Zilber’s celebrated quasiminimality conjecture states that a subset of $\mathbb{C}$ defined by polynomials and exponentials is either countable, or has countable complement. Koiran has asked whether the same could be true replacing $\exp$ with any unary entire function. We give a negative answer to Koiran’s question, using results from the theory of holomorphic approximation.