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.

A logarithm can be thought of as a function that stretches/squeezes along the number line. Given a series of values, taking the log will stretch the lower end of the range, and squash the upper end of the range - small values like 0.001, 0.01, and 0.1 that are "nearby" one another get stretched to cover a larger range of -3, -2, and -1, while large values like 1000, 10000, and 100000 that are "distant" from one another get squeezed to cover a smaller range of 3, 4, and 5.
When taking the log of a histogram, you're modifying distances between bars, but are doing so unevenly along the length of the histogram - you're stretching the left half of the histogram and squeezing the right half. Except in the case of very simple histograms with few values, this will always result in a different shape from what you started with. To preserve the shape, you'd need to shift or scale all values by the same amount, but the log effectively applies a scale factor that varies with the value being scaled.
