Is there any modality summary statistic? Is there any kind of (simple) summary statistic that would attempt to estimate the number of modes in a given sample? 
I mean, I would like to have some function $f$ that would behave as follows:
$$f(\textrm{observed unimodal data}) \approx 1$$
$$f(\textrm{observed bimodal data}) \approx 2$$
$$f(\textrm{observed N-modal data}) \approx N$$
etc. 
 A: Simple summary statistic? Not really. 
Any sort of summary statistic? Yes, but I'm guessing it's a harder problem than you may expect (and that the answers given by these methods are less reliable than you want). 
To help motivate the difficulty, consider the following histogram:

I believe looking at this plot, you would want your function to return 2 since there appears to be two clear modes. Note that this is the histogram built by R's hist function with default settings. 
But what if we decide to make the histogram a little finer? Now consider if we want 20 bins instead of 7 on the same dataset. 

Now maybe you want your function to tell you there are 4 peaks! What about 100 bins?

So many peaks! You can see that the number of modes in our histogram varies wildly with our choice of number of bins (btw: this is simulated data from four very distinct normals with 100 observations from each distribution. So the "true" answer is 4 modes. This would also be considered an easy example as the distributions are so distinct). 
While you may think this is just an obvious result of how histograms work, this is a very standard situation in these problems: you have some sort of smoothing parameter (in the histogram context, this would be the number of bins) and as you differ your smoothing parameter, you get wildly different number of reported modes. In general, deciding what value of the smoothing parameter to use is non-trivial. 
If you're still very interested, two topics to look into could be kernel density 
estimation and Gaussian mixture modeling. But be warned that you should not expect to be able to reliably estimate the true number of modes in the population!
