  Let $w$ be an unbounded radial weight on the complex plane. We study the
following approximation problem: find a proper holomorphic map $f:
\mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give
several characterizations of those $w$ for which the problem is solvable. In
particular, a constructive characterization is given in terms of tropical power
series. Moreover, the following natural objects and properties are involved:
essential weights on the complex plane, approximation by power series with
positive coefficients, approximation by the maximum of a holomorphic function
modulus. Extensions to several complex variables and approximation by harmonic
maps are also considered.
