For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when $p\to 0$, i.e., the geometric mean.
Furthermore, when $p,q$ are two real numbers, define $L_{p,q}(x,y)$ as the obvious fixed point satisfying the equation $$L_{p,q}(x,y) = L_{p,q}(M_p(x,y),M_q(x,y))$$ —meaning that $L_{p,q}(x,y)$ is defined as the common limit of the sequences $(x_n)$ and $(y_n)$ such that $(x_0,y_0) = (x,y)$ and $(x_{n+1},y_{n+1}) = (M_p(x,y),M_q(x,y))$. (Showing convergence is easy, and evidently, $L_{p,q} = L_{q,p}$.)
Thus, $L_{0,1}(x,y)$ is the famous arithmetic-geometric mean.
Among other things, all these functions satisfy $\min(x,y) \leq F(x,y) \leq \max(x,y)$, as well as $F(y,x) = F(x,y)$ and $F(\lambda x, \lambda y) = \lambda F(x,y)$ — so we could just define them from $F(1,x)$, which is continuous and monotonically increasing with value $1$ at $1$. It is well-known that $M_p(x,y) < M_q(x,y)$ whenever $x\neq y$ and $p<q$, so of course $M_p(x,y) < L_{p,q}(x,y) < M_q(x,y)$ under those circumstances.
Surprisingly, I can find no reference to the $L_{p,q}$ in the literature except for $(p,q)=(0,1)$ (mentioned above), and $(p,q)=(-1,1)$ which this page points out is just $M_0$. More generally, we have $$L_{p,-p}(x,y) = M_0(x,y)$$ and $$L_{0,p}(x,y) = \big(L_{0,1}(x^p,y^p)\big)^{1/p}$$ —both are easy, and $L_{p,p}(x,x) = M_p(x,y)$ is trivial.
I couldn't find anything intelligent to say about $L_{1,2}$ (the "arithmetic-quadratic mean"), however, let alone the other $L_{p,q}$. Experimentally, it satisfies $L_{1,2}(x,y) \leq M_{3/2}(x,y)$, something which is probably easy to prove and not interesting enough to try, so that's not my question. Instead, my question will be:
Can $L_{1,2}$, or more generally $L_{p,q}$ (with $0<p<q$), be expressed in closed form using elementary functions and the arithmetic-geometric mean $L_{0,1}$? Or at least using "standard" special functions (e.g. elliptic integrals)?
(I realize that the answer "no" might be exceedingly difficult to prove, so I'll be liberal in what I accept as an answer: perhaps the question should really be "please say something interesting about $L_{1,2}$ or $L_{p,q}$ in general".)
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…