To solve $f \circ g \circ h(x)$, we need to evaluate the composition of the functions in the order they appear, starting with the innermost function and proceeding outward. Let's go through the steps:

1. First, calculate $h(x)$: $h(x) = \sqrt{x}$

2. Next, find $g(h(x))$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 7$

3. Finally, calculate $f(g(h(x)))$: $f(g(h(x))) = f(\sqrt{x} - 7) = (\sqrt{x} - 7)^4 + 5$

Thus, $f \circ g \circ h(x) = (\sqrt{x} - 7)^4 + 5$.

Would you like to explore more about function compositions or have any other questions about this problem? Here are some related topics to consider:

1. What happens if you change the order of composition?
2. How does the graph of $f \circ g \circ h(x)$ compare to the individual graphs of $f(x)$, $g(x)$, and $h(x)$?
3. Can you derive a general method for simplifying nested function compositions?
4. What are the domain considerations for the composition $f \circ g \circ h(x)$?
5. How does modifying the functions $f$, $g$, or $h$ affect the final output?

Tip: When composing functions, always consider the domain restrictions imposed by each function, as this can impact the valid input values for the composition.