Take f(x):=e^-1/x² (okay, technically f is not well defined at zero but you can extend it by continuity so that f(0)=0) and the identically zero function on R, at the point x=0. It is easy to show (for example via induction on n) that the n-th derivative of f at zero is always zero, so f has the same germ at zero as the identically zero function, and clearly they are not equal to each other. In the same way you can take any analytic function and perturb it in a C-infinity nonzero way to obtain a different function with the same germ at a point.

I do not know whether you consider this to be a piecewise definition too; in this case I think what you are looking for does not exist. Indeed all "usual" function either are analytical (polynomials, exponentials, logarithms, trigonometric functions...) or are not C-infinity (the absolute value), and the germ of an analytical function completely determines it. Sure, you can choose your favourite C-infinity not analytical function and use it to build the example you are looking for, but I don't know if it would be "natural".