First, some notation:
$f^{\star0}(x)=x$
$f^{\star k}(x)=f\left(f^{\star k-1}(x)\right)$
So that for any integer $k$, $\log^{\star k}(x)=\underbrace{\log(\log(\dots(\log}_k(x))\dots))$.
I then came across the following integral:
$$\int\frac1{x\log(x)}dx=\log(\log(x))$$
Which I generalized into
$$\int\frac1{\prod_{n=0}^k\log^{\star n}(x)}dx=\log^{\star k+1}(x)$$
And was wondering if anyone had any idea as to how to extend this to non-integer $n$. This would be of interest to generalizing the tetration, where 'half iterations' of the log would come in handy.
A similar integral involving the exponential function:
$$\int\prod_{n=1}^k\exp^{\star n}(x)dx=\exp^{\star n}(x)$$
I tried to generalize it by letting $\frac d{dx}\log^{\star k+1}(x)=\frac1{\prod_{n=0}^\infty\log^{\star k-n}(x)}$, but then convergence is an issue. Same problem with the exponential case.
If someone manages a sneaky manipulation of sorts to generalize this, that'd be great.