Consider a Boolean circuit using (2-input) logical-and, (2-input) logical-or and logical-not as basic components. The depth of the Boolean circuit is the length of the longest path from the input to the output. I wonder if Boolean circuits with a depth Polylog in the number of inputs are sufficient to express any Boolean function. I only know that a depth of $O(n)$ is sufficient ($n$ is the number of inputs), by using the disjunctive normal form to construct a Boolean circuit.
Note that this question is different from the $NC$ complexity, since in $NC^i$ the size of the Boolean circuit is also constrained to be polynomial, while this question does not constrain the size. Thank you!