A crude approximation to $\pi$ using Gamma function

Question

Recently, I found using my calculator a weird result: $$ \label{eq:1}\tag{1} G_{\text{II}}=\Gamma(\Gamma(\pi +1)+1) \approx 7380\frac {5}{9}\text { or }\frac {66425}{9} $$ Here $\Gamma(z)$ is the gamma function. Hence, I began further experimentation, to see if this quantity could be expressed as $2n\pi+\theta$. Amazingly yes! $$ \label{eq:2}\tag{2} G_{\text{II}}\approx 2348\pi+4.096 $$ Using \eqref{eq:1} and \eqref{eq:2} we have, $$ \pi \approx \frac {1}{2348}(G_{\text{II}} - 4.096)=3.141592656560059\ldots, $$ and , $$ \pi \approx \frac {1}{2348}\left(\frac {66425}{9}-\frac {512}{125}\right)=3.141592655688056\ldots $$ The question is, how can I derive \eqref{eq:1} without using calculator? I mean, the Taylor series for the function $\Gamma(\Gamma(z+1)+1)$ is horrible: $$ 1+\gamma(\gamma-1)z+\frac {1}{12}(6\gamma^4-18\gamma^3+(\pi^2+6)\gamma^2-\pi^2\gamma +\pi^2 )z^2+\cdots $$ Here $\gamma$ is the Euler-Mascheroni constant. Any suggestions are welcome.