Sure, a simple example is the derivative of Riemann zeta for Re s < 1. Granted, usually you work with the logarithmic derivative which is a lot more tractable, but if for whatever reason you need the actual derivative, you're gonna have a lot of trouble with automatic differentiation, as the typical analytic continuation formulas involve some complicated improper integrals.

There are also functions that you only know by sampling (e.g. ocean temperatures) for which you assume smoothness. You need to pick an interpolation method, but sometimes you do not interpolate beyond the sampling points, because that's just making up numbers. When you're limited by your original sampling step size, you have little recurse but to compute derivatives by some finite differencing scheme.