What circuit depth is required to add?
To add two $n$-bit numbers, you need a circuit of depth $\Omega(\lg n)$.  Depth $O(\lg n)$ is also achievable, so the lower bound is tight.

There are many examples of such circuits; for instance, a carry lookahead adder has depth $O(\lg n)$ (and size $O(n)$).  One can even achieve depth $\lg n + o(\lg n)$; for instance, a Krapchenko adder has depth $\lg n + O(\lg^{1/2} n)$ (and size $O(n)$).

This all assumes bounded fan-in gates, as usual.  If you allow AND and OR gates with arbitrary fan-in, then you can achieve constant depth (depth 3, I think).  See, e.g., https://en.wikipedia.org/wiki/AC0.