A log log integral From numerical experiments I found that
$$
\int_0^1 \int_0^1 \log |\log |x-y|| dx dy = \log 2 - \gamma
$$
How can one prove it?
For more context, here is how I derived it non-rigorously. Starting from
$$
 \int_0^1 \int_0^1 |x-y|^n dx dy = \frac{2}{(n+1)(n+2)}
$$
Taking $k$-th derivative of $n$,
$$
 \int_0^1 \int_0^1 |x-y|^n(\log|x-y|)^k dxdy = (-1)^k 2k! \left( \frac{1}{(n+1)^{k+1}} - \frac{1}{(n+2)^{k+1}} \right)
$$
At $n=0$, we have
$$
 \int_0^1 \int_0^1 (\log |x-y|)^k dx dy = (-1)^k 2k!(1-2^{-(k+1)})
$$
Taking derivative of $k$ at zero, we have
$$
 \int_0^1 \int_0^1 \log \log |x-y| dx dy = \log 2 - \gamma + i\pi
$$
from which we obtain the integral in the question.
Remark. With the same argument, starting from
$$
\int_0^1 x^n dx = \frac{1}{n+1}
$$
we obtained the following result used in the accepted answer
$$
\int_0^1 \log |\log(x)| = -\gamma
$$
 A: Observe the symmetry about the line $y=x$. We have for instance
$$\begin{align*}
I &= \int_0^1 \int_0^1 \log|\log|x-y|| \, dx \, dy \\[1ex]
&= 2 \int_0^1 \int_y^1 \log(-\log(x-y)) \, dx \, dy
\end{align*}$$
Now,
$$\begin{align*}
I &= 2\int_0^1 \int_0^{1-y} \log(-\log(x)) \, dx \, dy \label{1}\tag{1} \\[1ex]
&= 2 \int_0^1 \int_0^{1-x} \log\left(\log\left(\frac1x\right)\right) \, dy \, dx \label{2}\tag{2} \\[1ex]
&= 2 \int_0^1 (1-x) \log\left(\log\left(\frac1x\right)\right) \, dx \\[1ex]
&= -2\gamma - 2\int_0^1 x \log\left(\log\left(\frac1x\right)\right) \, dx \label{3}\tag{3} \\[1ex]
&= -2\gamma - 2 \int_1^\infty \log(\log(x)) \, \frac{dx}{x^3} \label{4}\tag{4} \\[1ex]
&= -2\gamma - \int_0^\infty  (\log(x)-\log(2)) e^{-x} \, dx \label{5}\tag{5} \\[1ex]
&= -2\gamma + \gamma + \log(2) \int_0^\infty e^{-x} \, dx \label{6}\tag{6} \\[1ex]
&= \log(2) - \gamma
\end{align*}$$

*

*\eqref{1} : substitute $x\mapsto x+y$

*\eqref{2} : change integration order

*\eqref{3} : identity (5) in Mathworld entry on the Euler-Mascheroni constant.

*\eqref{4} : substitute $x\mapsto\frac1x$

*\eqref{5} : substitute $x\mapsto e^{-x/2}$

*\eqref{6} : identity (4) in Mathworld entry on the Euler-Mascheroni constant.

