Is it meaningful to take the derivative of a function a non-integer number of times?

To expand on Jonas's comment: Yes, it makes sense. For the case of the power function, one can consider

$$\frac{\Gamma(n+1)}{\Gamma(n-\alpha+1)}x^{n-\alpha}$$

as the $\alpha$-th derivative of the power function $x^n$, where $\Gamma(z)$ is the gamma function, the generalization of the factorial to the complex plane.

In general, one has a number of definitions for so-called "fractional derivatives", or, as Spanier and Oldham prefer to call it, the "differintegral". Negative values of $\alpha$ in expressions like the one given above correspond to integration, positive values correspond to differentiation, and in general $\alpha$ can be complex.

There's a lot of things to look at (Caputo derivatives, Riemann-Liouville integrals, Grunwald-Lednikov series), and I suggest you look at the book I linked to first, and then search around the web. Have fun!

Here's a blog post that motivates a definition of fractional derivatives in terms of Fourier transforms.