f(x) = e^(-1/x2) for x != 0, f(0) = 0. It's relatively easy to show this is infinitely differentiable at x=0 and every derivative is 0.

The intuition that an infinitely differentiable function can be described globally by its derivatives locally is actually true for complex differentiable functions, and this property is sometimes referred to as "rigidity" of complex-differentiable (or analytic/holomorphic) functions. It doesn't hold for functions that are only differentiable along the real axis.