The given expression is:

$\frac{1}{(-1)^{-(-1)}}$

Let's break it down step by step:

1. First, simplify the exponent $-(-1)$. This becomes: $-(-1) = 1$

2. Now the expression becomes: $\frac{1}{(-1)^1}$

3. $(-1)^1$ is simply $-1$, so the expression becomes: $\frac{1}{-1}$

4. $\frac{1}{-1} = -1$.

Thus, the value of the expression is:

$\boxed{-1}$

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How do negative exponents work in general?
2. What is the rule for simplifying powers of negative numbers?
3. What is the difference between $(-1)^n$ and $-1^n$?
4. How do reciprocal and negative exponents relate?
5. Can you apply these rules to a similar problem, like $\frac{1}{(-2)^{-2}}$?

Tip: Always remember that negative exponents represent the reciprocal of the base raised to the corresponding positive exponent.