Given Taylor series and enough data so as to not risk over-fitting, do you actually need to think about if your phenomenon is following an exponential, quadratic, logarithmic, ..., behaviour? I'm sure you can come up with counter-examples of when this isn't a good idea, but if we're very general and pragmatic, is it usually 'fine' to just fit the data to some degree nine polynomial hoping whatever bizzare pattern is hiding in the data will be very well approximated by its power series?
Imagine replacing an arbitrary model with polynomial parameters with a series of dummies for all values of the explanatory variables and all their interactions. If you have enough experimental data that's going to be as general as possible. The highest order polynomial function of all the interactions is going to fit the data as well. So if you have the data and setting to estimate the fully saturated model that will be model free.
For example, consider that the true model is where X and Y take dummy values only:
Z = F(X,Y)
You could also write this model as:
Z = beta0 + beta1 * 1(X) + beta2 * 1(Y) + beta3 * 1(X) * 1(Y)
These functions match everywhere on the values taken by X and Y. This gets to be quite thorny if you have multiple parameters and they take multiple values, but the principle is the same.
I learned about this from Mostly Harmless Econometrics (p. 48 - 51) where they argue that the case of saturated modeling implies that linear modeling is equally general as non-linear modeling. Moving into non-saturated models means that non linear models can cover functions with fewer parameters, but with enough free parameters and the data to estimate them, they cover the same set of models.