Problem 808

Solution:

Consider that $f(x) = x^{2} + 6x + 7 = (x+3)^{2} - 2$. So $f(x) \geq -2$ for real numbers $x$. Also, $f$ is increasing on the interval $[-3, \infty)$.

Therefore
$$\begin{gathered} f(f(x)) \geq f(-2) = -1, \\ f(f(f(x))) \geq f(-1) = 2, \end{gathered}$$
and
$$f(f(f(f(x)))) \geq f(2) = 23$$
Thus, the minimum value of $f(f(f(f(x))))$ is $23$ and equality is obtained when $x = -3$.