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Yes I'm sure it roughly follows a power law within some scale range. Of course it depends on the beach. One problem is that the sea often separates stones by size, so small stones get washed a different distance to large stones. Brighton beaches (England) tend to have stones all about the same size.
I'm sure I've skimmed a few articles which give a basis for the power law, I don't have them with me but the book Fractals, Chaos, Power Laws would I think be a good place to look https://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/0486472043.

There's a really good point here, that you need to get the dimension to actually quantify the beach. Counting stones, weighing total mass or summing stone surface area are all dependent on how fine grained you choose to go (see https://en.wikipedia.org/wiki/Coastline_paradox). By contrast, if you work out the fractal dimension and obtain the fractal measure (a generalisation of count, length, area... to fractional dimensions), this is invariant to how fine grained you go. You get a real measurement to compare different beaches of the same power law dimension.

By the way, the fractal shape in your image is a type of 'cluster-sponge' in my fractal classification scheme https://sites.google.com/site/simplextable.

please post this in v2 too!
(or I'll just copy/post this over there if you're ok)