Symmetric functions in NC¹
We can treat the input $x$ as a boolean array. Using an $O(\log n)$ depth sorting network (such as the AKS network), we can sort $x$ in nonincreasing order. Calling the result $y$, using $O(1)$ more depth we can compute $z_i = y_i \land \lnot y_{i+1}$ (extended in the right way to the boundary). The vector $z_0,\ldots,z_n$ is the indicator vector of the Hamming weight of $x$. At this point you can compute any symmetric function using $O(\log n)$ more depth.
This shows that there is a universal constant $C$ such that all symmetric functions can be computed in depth $C\log n$ using constant fan-in circuits.