I notice both wikipedia and mathworld have the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ that:
$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx=\sum\limits_{n=0}^m\dfrac{(-1)^n(n+1)^{n-1}}{n!}\Gamma(n+1,-\ln x)+\sum\limits_{n=m+1}^\infty(-1)^na_{mn}\Gamma(n+1,-\ln x)$ , where $a_{mn}=\begin{cases}1&\text{if}~n=0\\\dfrac{1}{n!}&\text{if}~m=1\\\dfrac{1}{n}\sum\limits_{j=1}^nja_{m,n-j}a_{m-1,j-1}&\text{otherwise} \end{cases}$
How does this result derived?
Somebody posted a nice question on sci.math long time ago about finding the expansion of the function exp(exp(x)) in terms of x. Two people answered correctly: Robert Israel and Leroy Quet. I used Leroy's result and found that a secondary pattern emerged, which is the series up to m, which is related to the W function, for which I later published (this and some other results as incidental by indcution). In reality Leroy's name should be on the reference, since he solved the basic step of finding the first expansion. At the time I underestimated the importance of his result and forgot to give him credit for it.