The problem asks for the limit:

$\lim_{x \to -1} f(f(x))$

Step 1: Evaluate $f(x)$ as $x \to -1$

From the graph, we observe that as $x \to -1$, the value of $f(x)$ approaches 3. Therefore,

$f(-1) = 3$

Step 2: Now find $f(f(x))$ at $f(x) = 3$

We already know that $f(-1) = 3$, so now we need to determine $f(3)$ from the graph. From the graph, we can see that:

$f(3) = 1$

Conclusion:

$\lim_{x \to -1} f(f(x)) = f(3) = 1$

Thus, the limit is:

$\boxed{1}$

Let me know if you'd like more details or further explanation. Here are some related questions you can explore:

1. How do you determine the limit from a graph?
2. What are the conditions for a limit to exist at a point?
3. What is the difference between one-sided limits and two-sided limits?
4. How can we apply the chain rule to composite functions like $f(f(x))$?
5. How does continuity affect the calculation of limits?

Tip: When evaluating limits from graphs, always check for any discontinuities or jumps at the point you're analyzing.