Let $z$ be a nonreal complex number.  Find the smallest possible value of
\[\frac{\text{Im}(z^5)}{[\text{Im}(z)]^5}.\]Note: For a complex number $z,$ $\text{Im}(z)$ denotes the imaginary part of $z.$
Let $z = x + yi,$ where $x$ and $y$ be real numbers.  Since $z$ is nonreal, $y \neq 0.$

Now,
\[z^5 = (x + yi)^5 = x^5 + 5ix^4 y - 10x^3 y^2 - 10ix^2 y^3 + 5xy^4 + iy^5,\]so
\[\text{Im}(z^5) = 5x^4 y - 10x^2 y^3 + y^5.\]Hence,
\begin{align*}
\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{align*}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 $\boxed{-4}.$