Answer
Does not exist.
Work Step by Step
Suppose, we have a vector field $G$ such that $div [curl (G)]=0$
Let us consider $F=ai+b j+c k$
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
Given: $curl G=xi+yj+zk $
Here, we have $div[curl(G)]=div (xi+yj+zk) $
and $div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}=\dfrac{\partial (x)}{\partial x}+\dfrac{\partial (y)}{\partial y}+\dfrac{\partial (z)}{\partial z}$
$\implies div[curl(G)]=1+1+1=3$
Hence, we can conclude that there does not exist such a vector field $G$.