Step 1: Plan:
Test each option in the equation that the nested radical must satisfy.
Step 2: Equation:
Let the value be $x$. Then $x^2 = -1 - x$, i.e. $x^2 + x + 1 = 0$. Multiplying by $(x-1)$ gives $x^3 = 1$, so $x$ is a cube root of unity other than $1$.
Step 3: Pick from the list:
The value $1$ is excluded since it is the real cube root. $-1$ cubes to $-1$. $-\omega$ cubes to $-1$. Only $\omega^2$ cubes to $1$ and is non-real. So the value is $\omega^2$.
Final Answer:
The value of $x$ is $\omega^2$, option (D).
\[ \boxed{\omega^2} \]