Is there a name for a distribution where I can take log of its histogram and get back the same histogram? I am imagining some data distribution where taking the log over and over again yields a distribution that has the exact same shape.

To put this in more mathematical form, you want to be able to start with a random variable $X$, take a logarithm, then perhaps add and multiply by some numbers and get back the same distribution. $Y=a+b\log X$ and $X$ have the same distribution.
This is possible for a distribution that puts all its probability on two points. The logarithm puts all the probability on two different points, and you can rescale it back to the original distribution.
It's not possible otherwise. Let $x_1$ and $x_2$ be two values of $X$ and consider the point halfway between them: $x_m=\frac{1}{2}(x_1+x_2)$. Because $\log$ is a concave function, $\log x_m$ will always be less than $\frac{1}{2}(\log x_1+\log x_2)$.  If we write $y_1$ and $y_2$ for the transformed values of $x_1$ and $x_2$, we can choose the scaling so that $y_1=x_1$ and $y_2=x_2$. But $y_m<x_m$, rather than $y_m=x_m$. So we can only get the same shape back if there is no probability at $x_m$.  Basically the same argument works for a point 1/3 of the way between $x_1$ and $x_2$, or any other intermediate position.
