Is log transforming square root transformed data a legitimate data transformation? Is it legitimate to do a "double transformation" on data? Specifically, log transforming data which has already been square root transformed, or conversely, square root transforming log transformed data? Does this even make sense?
 A: Specifically, no. 
Using natural logarithms for concreteness, and a generic variable $x$, the transformation $\ln \sqrt{x}$ is defined easily for positive numbers, but also can be seen to be just $\ln x^{0.5} = 0.5 \ln x$. Hence it has exactly the same effect in terms of effect on nonlinearity, asymmetry, lack of Gaussianity -- or whatever the transformation is designed to cure or improve -- as $\ln x$. 
The transformation $\sqrt{\ln x}$ is defined easily for $x > 1$, but that seems also to be an artificial constraint. I'd be happy to learn of a rationale but I can't think of one. If you want a transformation that is stronger than the logarithm, the reciprocal is a simpler transformation to consider first. Reciprocals can be applied usefully so long as all values are positive or all values are negative. 
Generally, there certainly are useful transformations that can be regarded as two-step functions, e.g. the logit $\ln [p / (1 - p)]$ or the angular transformation $\arcsin \sqrt{p}$. Here the domain is now proportions or probabilities $p$ and $0 < p < 1$ for logit and $0 \le p \le 1$ for angular. If a transformation is useful and natural enough, we tend to regard it as a function directly defined and named, so that (e.g.) statistical software commonly supplies logit functions for immediate use. 
Some might want to underline that in some contexts complex results, i.e. those with real and imaginary parts, might be interesting or useful. 
A: By the properties of logarithms, $$
y = \sqrt{x}
$$$$
\log(y)=0.5\log(x)
$$
So the net effect in something like a regression setting is to add a constant multiplier out front of the variable, i.e. to double the coefficient value of the log-square-root compared to simply taking the log. 
But in other contexts, the effect won't matter at all! If you're using decision tree methods (e.g. random forest), and x > 0, then it won't even matter because these models are insensitive to monotonic transformations of the data.
