Acceleration *can jump* from zero to something. When it does, its derivative is not defined, so the series of position derivatives stops after the second one.

From the question: "A change is velocity is acceleration, so the value of the acceleration would have to increase from zero to some value."

The first part is true, we need non-zero acceleration. The second part is misleadingly phrased. It can be read as "acceleration has to increase continuously", which is false. In contrast to velocity, acceleration can jump from zero to something.

Philosophically speaking, there is no law that says "in physics, derivatives are always smooth". Some derivatives just jump.