To simplify $\left( \left( \frac{1}{6} \right)^3 \right)^3$, we apply the Power Rule of Exponents, which states that $(a^m)^n = a^{m \cdot n}$.

Here, the base is $\frac{1}{6}$, and we have:

$\left( \left( \frac{1}{6} \right)^3 \right)^3 = \left( \frac{1}{6} \right)^{3 \cdot 3} = \left( \frac{1}{6} \right)^9$

So, the exponent is 9.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Simplify $\left( \frac{2}{5} \right)^4$.
2. Apply the power rule to simplify $\left( 3^2 \right)^4$.
3. What is the result of $\left( 7^{-2} \right)^3$?
4. If $\left( x^4 \right)^5 = x^k$, what is $k$?
5. Simplify $\left( \frac{3}{8} \right)^2$.

Tip: When using the power rule, always multiply the exponents if both are applied to the same base.