Asymptotics of $\operatorname{agm}(1,x)$ when $x\to\infty$
For every $x>1$, $$\mathrm{agm}(1,x)=x\cdot\mathrm{agm}(1,x^{-1})=\frac{\pi x}{2K(u(x))}$$ where $$u(x)^2=1-x^{-2}$$ and $K$ denotes the complete elliptic integral of the first kind.
When $x\to\infty$, $u(x)\to1$. The asymptotic expansion of $K(k)$ when $k\to1$ reads $$K(k)=-\frac12\log|1-k|+O(1)$$ hence, when $x\to\infty$, $$\mathrm{agm}(1,x)=\frac{\pi x}{-\log|1-u(x)|+O(1)}=\frac{\pi x}{2\log x+O(1)}$$ in particular,
$$\lim_{x\to\infty}\frac{\log x}x\cdot\mathrm{agm}(1,x)=\frac\pi2$$
The fundamental asymptotic here is $$\frac{\pi} {2\operatorname {agm} (1,k)}=K(k') = \log\frac{4}{k}+o(1)$$ as $k\to 0^{+}$. This can be written as $$\frac{\pi} {2k\operatorname {agm} (1,1/k)}=\log\frac{4}{k}+o(1)$$ Putting $x=1/k$ we see that $$\frac{\pi x} {2\operatorname {agm} (1,x)}=\log 4x+o(1)$$ as $x\to\infty $. This implies that your function $\operatorname {agm}(1,x)$ behaves like $\pi x/\log 16x^{2}$ as $x\to\infty$.
Another way to look at this is in terms of nome $q=e^{-\pi K'/K} $ and Jacobi's theta functions. Note that $$k=\frac{\vartheta_{2}^{2}(q)}{\vartheta_{3}^{2}(q)}=\frac{4\sqrt{q}\psi^{2}(q^{2})}{2K/\pi}$$ where $\psi$ is one of Ramanujan's theta functions. When $q\to 0^{+}$ then $2K/\pi\to 1,\psi(q^{2})\to 1$ so that $k/4\sqrt{q}\to 1$. And since $\pi K'/K=\log q^{-1}$ it follows that $K'/\log \sqrt{q} \to - 1$. And this combined with $k/4\sqrt{q}\to 1$ gives us $K'/\log(4/k)\to 1$ as $k\to 0$.
You may also obtain the asymptotic for $K'$ by noting that both $K(k) $ and $K(k') $ satisfy the same differential equation and an analysis of indicial equation gives the link between $K$ and $K'$ namely $$K(k') =\frac{2K(k)} {\pi}\log\frac{4}{k} -2\left\{\left(\frac{1}{2}\right)^{2}\left(\frac{1}{1\cdot 2}\right)k^{2}+\left(\frac{1\cdot 3}{2\cdot 4}\right)^{2}\left(\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}\right)k^{4}+\dots\right\}$$
It is of interest to note that Ramanujan presented the above asymptotic in the following grand form $$\lim_{x\to 0^{+}}\frac{1}{x}\exp\left(-\pi\cdot\dfrac{{}_{2}F_{1}\left(\dfrac{1}{2},\dfrac{1}{2};1;1-x\right)}{{}_{2}F_{1}\left(\dfrac{1}{2},\dfrac {1}{2};1;x\right)}\right)=\frac{1}{16}$$ and proved it using certain identities relating hypergeometric functions. Using this formula and some more hypergeometric identities (mainly Gauss-Landen quadratic transformation) Ramanujan inverted the relation $$q=\exp\left(-\pi\cdot\dfrac{{}_{2}F_{1}\left(\dfrac{1}{2},\dfrac{1}{2};1;1-x\right)}{{}_{2}F_{1}\left(\dfrac{1}{2},\dfrac {1}{2};1;x\right)}\right)$$ to get $$x=\frac{\vartheta_{2}^{4}(q)} {\vartheta_{3}^{4}(q)}, \, {}_{2}F_{1}\left(\frac{1}{2},\frac {1}{2};1;x\right)=\vartheta_{3}^{2}(q)$$ and further deduced the transformation formula for theta functions. This is quite unlike the modern approaches (based on Poisson summation formula) and Jacobi's approach (based on integral transformations). Ramanujan's approach is more powerful and it led him to consider expressions $$q_{r}(x) = \exp\left(-\frac{\pi} {\sin(\pi/r) } \cdot\dfrac{{}_{2}F_{1}\left(\dfrac{1}{r},\dfrac{r-1}{r};1;1-x\right)}{{}_{2}F_{1}\left(\dfrac{1}{r},\dfrac {r-1}{r};1;x\right)}\right)$$ for $r=3,4,6$ and develop alternative theories of theta functions.