Calculate the Mean mean of two numbers
Wolfram Language (Mathematica), 52 bytes

#//.x_:&gt;N@{M@x,E^M@Log@x,1/M[1/x],M[x^2]^.5}&amp;
M=Mean


Try it online!

In my first approach I used these builtins
Mean GeometricMean  HarmonicMean and RootMeanSquare  

Here are some substitutions for saving bytes  

HarmonicMean -->  1/Mean[1/x]  by @Robin Ryder (3 bytes saved)
GeometricMean --> E^Mean@Log@x by @A. Rex (2 bytes saved)
RootMeanSquare --> Mean[x^2]^.5 by @A. Rex (4 bytes saved)   

finally we can assign Mean to M (as proposed by @ovs) and save 5 more bytes