Negative Log inside Negative Log Is there a name for a function which is like $f(x)=-\log(-\log(x))$ where $0<x<1$?
Or, is there any name for this function $g(x)=x+\log(\frac{1}{x})$ where $0<x$?
Exchanging $x=-\log k$ in $g(x)$ gives $f(k)$ and I would like to know about those functions any deeper, but I am having trouble searching about them. If there are any specific name or related function that I can search for, I will be very glad to know.

Your first function corresponds to the inverse of the standard cumulative Gumbel distribution and occurs in extreme value statistics. In statistics, the inverse of a cumulative distribution is also called a quantile function. So, that makes it the standard Gumbel quantile function.
Other than that, I cannot see a use for devoting special attention to this function, in any case not at the analysis level, as it is just a composite function and the interesting function from that point of view is just the logarithm.
