We are given the function $f(x) = \sqrt{x - \sqrt{x - \sqrt{x - \sqrt{x - \dotsb}}}}$, and we need to find the largest three-digit value of $x$ such that $f(x)$ is an integer.

Step 1: Understanding the Recursion

Let $y = f(x)$. The function $f(x)$ represents an infinite nested radical, so we can express it as: $y = \sqrt{x - y}.$ We square both sides of this equation to eliminate the square root: $y^2 = x - y.$ Rearranging terms, we get: $y^2 + y = x.$ Thus, the value of $x$ can be written as: $x = y^2 + y.$ This equation gives us the value of $x$ in terms of $y$.

Step 2: Finding Integer Solutions for $x$

We are looking for the largest three-digit value of $x$ such that $f(x)$ is an integer. To find integer values of $x$, we need $y$ to be an integer. The expression for $x$ is: $x = y^2 + y = y(y + 1).$ This represents the product of two consecutive integers, $y$ and $y + 1$.

Step 3: Maximizing $x$

To find the largest three-digit value of $x$, we need to determine the largest integer $y$ such that $x = y(y + 1)$ is a three-digit number. We solve the inequality: $y(y + 1) \leq 999.$ First, approximate $y(y + 1) \approx y^2$, so we solve: $y^2 \leq 999.$ Taking the square root of both sides: $y \leq \sqrt{999} \approx 31.6.$ Thus, the largest integer $y$ is 31. We now compute $x$ for $y = 31$: $x = 31 \times 32 = 992.$

Step 4: Verifying the Value of $f(x)$

For $x = 992$, we check that $f(x)$ is an integer by solving the equation $y^2 + y = 992$. Substituting $y = 31$: $31^2 + 31 = 961 + 31 = 992,$ which is correct. Thus, $f(992) = 31$, which is an integer.

Step 5: Conclusion

The largest three-digit value of $x$ such that $f(x)$ is an integer is $\boxed{992}$.

---

If you'd like more details or have any questions, feel free to ask! Here are some related questions to explore further:

1. How would the function $f(x)$ change if we replaced the square root with a different operation, such as a cube root?
2. What would happen if we replaced $f(x) = \sqrt{x - \sqrt{x - \sqrt{x - \dotsb}}}$ with a finite nested radical, say 4 levels of square roots?
3. Can we derive a general formula for the value of $x$ in terms of $y$ for any number of nested square roots?
4. What are some other examples of equations that involve infinite nested radicals and how to solve them?
5. How do we know that the expression $y^2 + y$ will always be an integer for integer values of $y$?

Tip: When dealing with infinite nested radicals, converting the problem into a recursive equation like $y = \sqrt{x - y}$ is often a helpful first step.