Here is the motivation of this question: $\mathbb C^n$ is already "local" in algebraic category. In other words, algebraic subvarieties of $\mathbb C^n$ are affine, so they are common zero locus of finitely many polynomials defined on $\mathbb C^n$.
However, $\mathbb C^n$ is not "local" enough for analytic varieties, since by definition (Griffiths & Harris, page 12), an analytic subvariety $X$ of $\mathbb C^n$ is that for each $x\in X$, there is an open neighborhood $U$ of $x$ in $\mathbb C^n$, such that $X\cap U$ is common zero loci of holomorphic functions $f_1,...,f_k$ defined on $U$.
Of course, globally defined holomorphic functions will do the job, for example, the graph of the entire function $z\mapsto e^z$ produces the simplest analytic subvariety of $\mathbb C^2$ that is not algebraic. However, is there an example of a closed analytic subvariety of $\mathbb C^n$ not defined by global holomorphic functions on $\mathbb C^n$?