An example I like is $\exp(-\frac{1}{x^2})$ and the "bump functions" one can construct with it.

First of all, this example is important in differential geometry (e.g. Whitney's embedding theorem) and complex analysis (as an example of a real $C^\infty$ function which isn't holomorphic).

In second place, even in first year calculus it's an important illustration of the concept of derivative and of Taylor's theorem. It's important in my opinion to understand why all derivatives at zero are zero (i.e. because it goes to zero faster then any polinomial) but even so the function is changing values.