This is a complex nested fraction problem. Let us evaluate the value of the expression step by step.

The given expression:

$1 - \frac{1}{1 - \frac{1}{1 - \frac{1}{1 - \frac{1}{1 - \frac{2}{3}}}}}$

Step 1: Start from the innermost fraction

The innermost part is: $1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}.$

Step 2: Substitute and simplify the next level

The next fraction becomes: $1 - \frac{1}{\frac{1}{3}} = 1 - 3.$ $1 - 3 = -2.$

Step 3: Substitute $-2$ into the next level

$1 - \frac{1}{-2} = 1 - (-\frac{1}{2}).$ $1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}.$

Step 4: Substitute $\frac{3}{2}$ into the next level

$1 - \frac{1}{\frac{3}{2}} = 1 - \frac{2}{3}.$ $1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}.$

Step 5: Substitute $\frac{1}{3}$ into the final level

$1 - \frac{1}{\frac{1}{3}} = 1 - 3.$ $1 - 3 = -2.$

Final Answer:

$\boxed{-2}$

Let me know if you'd like a detailed explanation for any step or have further questions! Here are some follow-up questions for practice:

1. How do nested fractions simplify when the denominators involve negative values?
2. What happens if the innermost fraction is zero?
3. How can we generalize this process for larger nested fractions?
4. Why does simplifying from the innermost fraction outward make the calculation easier?
5. Can you create a similar problem with different numbers for practice?

Tip: Always simplify from the innermost part of a nested fraction to avoid errors.