Given a function and its complexity, what is the complexity of its circuit?
There is no mapping from algorithmic complexity to circuit size. A counterexample: consider the language of strings of all ones, $1^n$, for only those $n$ such that the $n$-th Turing Machine rejects the string $1^n$ within $2^{2^n}$ time. Then this function requires $\Omega\left(2^{2^n}\right)$ time to compute on any machine, but the circuit for it only has size $O(n)$, because you can "hard-wire" the answer into the circuit, which then only checks that all the inputs are $1$.

D.W. provides a construction from an algorithm to a circuit. However, note that this gives only an upper bound, which is not necessarily tight. In particular, no function requires circuits of size larger than $O\left(\frac{2^n}{n}\right)$, because that is the size of a circuit which simply enumerates that function's truth table for inputs of size $n$.

Consequently, if an algorithm takes $O(2^{2^n})$ time, then you can use D.W.'s construction for a circuit of size $O(2^{2^n}\log(n)^2)$, but it is immediate that this bound is not tight; there is also a circuit of size $O(\frac{2^n}{n})$, and there may be a much smaller circuit for it, as in the example above. What we really want to know when we ask what is the size of this function's circuit is a lower bound: a minimum size for any circuit.

Circuit size lower bounds is a particularly enigmatic part of computational complexity about which very little is known. For example, we do not know whether or not SAT can be solved with $O(n)$-size circuits, and we have not ruled out that each problem in NEXP can be solved with polynomial-size circuits.