1. **State the problem:** Simplify the nested fraction expression: $$\frac{1}{1 - \frac{1}{1 - \frac{1}{x}}}$$ 2. **Understand the structure:** This is a complex fraction with nested denominators. We will simplify from the innermost fraction outward. 3. **Start with the innermost denominator:** $$1 - \frac{1}{x} = \frac{x - 1}{x}$$ 4. **Replace the innermost denominator:** $$\frac{1}{1 - \frac{1}{x}} = \frac{1}{\frac{x - 1}{x}} = \frac{1}{1} \times \frac{x}{x - 1} = \frac{x}{x - 1}$$ 5. **Now simplify the next denominator:** $$1 - \frac{1}{1 - \frac{1}{x}} = 1 - \frac{x}{x - 1} = \frac{(x - 1)}{(x - 1)} - \frac{x}{x - 1} = \frac{x - 1 - x}{x - 1} = \frac{-1}{x - 1}$$ 6. **Substitute back into the original expression:** $$\frac{1}{1 - \frac{1}{1 - \frac{1}{x}}} = \frac{1}{\frac{-1}{x - 1}} = 1 \times \frac{x - 1}{\cancel{-1}} \times \cancel{-1} = -(x - 1)$$ 7. **Final simplified expression:** $$-(x - 1) = -x + 1$$ **Answer:** $$\boxed{-x + 1}$$