Let $x,y,z$ be positive real numbers. And let $\text{AM}$, $\text{GM}$, $\text{HM}$ respectively be the arithmetic mean, geometric mean, and harmonic mean.
Define $$a_n=\text{AM}(a_{n-1},g_{n-1},h_{n-1}),$$ $$g_n=\text{GM}(a_{n-1},g_{n-1},h_{n-1}),$$ $$h_n=\text{HM}(a_{n-1},g_{n-1},h_{n-1})$$
with $a_0=x,$ $g_0=y,$ $h_0=z$.
I found that $$\lim\limits_{n\to\infty} a_n=\lim\limits_{n\to\infty} g_n=\lim\limits_{n\to\infty} h_n$$
This is an analog of other mean combinations such as the arithmetic-geometric mean and geometric-harmonic mean. The arithmetic-harmonic mean is just the geometric mean. Therefore this would be considered the arithmetic-geometric-harmonic mean (AGHM).
Indeed, an analogous identity as seen with the GHM seems to hold: $$\text{AGHM}(x,y,z)=\frac{1}{\text{AGHM}(\frac1x,\frac1y,\frac1z)}$$
But how do we prove these things? Maybe unlikely, but can we find a closed form for the AGHM? Could the AGHM have any application?
Interesting: I found from experimentation that $\text{AGHM}(x,y,\sqrt{xy})=\sqrt{xy}$.