Arithmetic-quadratic mean and other "means by limits of means"

These type of questions are investigated in "Compromise, consensus, and the iteration of means" and "Markov chains, Gauss soups, and compromise dynamics" by Ulli Krause. These papers also provide more pointers to relevant literature.

An earlier reference is "Pi and the AGM" by Borwein and Borwein and Chapter 8 there introduces the concept of limits of iterated means (only for two different means, as far as I see). A particular results is: If $M$ and $N$ are means, and the iterated mean converges to a function $\Phi$, then this is again a mean and characterized by $$\Phi(M(a,b),N(a,b)) = \Phi(a,b).$$

So the function you are looking for is simply characterized by the functional equation $$L_{1,2}(1,x) = L_{1,2}\left(\sqrt{\tfrac{1+x^2}2},\tfrac{1+x}2\right),$$ but you seem to know that already. The book contains a few examples, but I could not find an example in the direction you are asking…