Not really. Sure, the technical term is an “analytic function”, the definition of which is, essentially, “any function that is described by it’s Taylor series”. The surprising thing is how many functions are analytic. All polynomials. The exponential, and sums of exponentials. The trigonometric functions. Bessel functions. All complex, differentiable functions. Combinations of the above.

Nearly any function you are likely to encounter in a physics class, to the best of my knowledge, is analytic (IANAP).

Another way to look at this is: “almost all” smooth functions are “nearly” polynomials. Why should trigonometric functions be “nearly” polynomials? Why should exponentials? Why should there be any connection at all between the trig functions and the polynomials? I still say this is a surprising result.