jmc

Let $z=x+yi,$ where $x$ and $y$ be real numbers. Since $z$ is nonreal, $y\ne 0.$

Now, $$z^5=(x+yi{)}^5=x^5+5i{x}^{4}y-10x^3{y}^2-10i{x}^2{y}^3+5x{y}^4+i{y}^5,$$so $$\text{Im}(z^5)=5x^{4}y-10x^2{y}^3+y^5.$$Hence, $$\begin{array}{rl}\frac{\text{Im}(z^5)}{[\text{Im}(z){]}^5}& =\frac{5x^{4}y-10x^2{y}^3+y^5}{y^5}\\ & =\frac{5x^4-10x^2{y}^2+y^4}{y^4}\\ & =5\cdot \frac{x^4}{y^4}-10\cdot \frac{x^2}{y^2}+1\\ & =5t^2-10t+1,\end{array}$$where $t=\frac{x^2}{y^2}.$ Now, $$5t^2-10t+1=(5t^2-10t+5)-4=5(t-1{)}^2-4\ge -4.$$Equality occurs when $t=1,$ which occurs for $z=1+i,$ for example. Therefore, the smallest possible value is $\boxed{-4}.$