Bounding or evaluating an integral limit I don't know if this integral was in the literature, then refers it please. This question is a curiosity after I did some experiments with Wolfram Alpha online calculator, and I hope that has mathematical meaning, that is, that I believe that it is well-defined.
Consider the sequence of integrals with first term
$$\int_1^{\infty}x^{-x}dx,$$ 
and second 
$$\int_1^{\infty}\int_1^{\infty}x^{-y}y^{-x}dxdy.$$ 
And now the third term of  this sequence being a triple integral, with factors in this new integrand $x^{-y}, y^{-x}$ and $z^{-x}, z^{-y}$, and also $z^{-x}$ and $z^{-y}$.
Thus I am saying this integral $$\int_1^{\infty}\int_1^{\infty}\int_1^{\infty}x^{-y-z}y^{-x-z}z^{-x-y}dxdydz.$$ And the next integral of our infinite sequence of integrals, with a similar pattern than previous. 

Question. Is it known what is the value of the limit integral? If you know the literature please refers it. If it isn't in the literature, can you then calculate an approximation of such limit integral? Or well a good upper bound. Many thanks.


Conjecture: they tend to zero:
$\int_1^{\infty } x^{-x} \, dx\approx 0.70417$
$\int _1^{\infty }\int _1^{\infty }x^{-y} y^{-x}dydx\approx 1.2259$
$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-y-z} y^{-x-z} z^{-x-y}dzdydx\approx 0.120842$
$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-w-y-z} y^{-w-x-z} z^{-w-x-y} w^{-x-y-z}dwdzdydx\approx 0.0116407$
