PDF with two exponents I found empirically a density function that becomes linear when one takes logarithm two times. So, the density function is something of the form
$$\alpha e^{\beta e^{-\gamma x}}.$$
I cannot find out whether it is something well-known or not. Does this distribution appear in any applied context?
 A: Looks like the Gumbel (https://en.wikipedia.org/wiki/Gumbel_distribution) distribution, or something related to it.
A: I agree it looks like a Gumbel distribution, and extreme value distribution, but here is a thing. What is the support of X such as $f_X(x) = \alpha e^{\beta e^{-\gamma x}}$ is well-defined?
Then, if you found, then since it is a continuous random variable, you can compute by taking integral from the support. For example, for integration convenience, we take x is defined from $-\infty$ to $\infty$, then I try
$$\int_{-\infty}^\infty \alpha e^{\beta e^{-\gamma x}}dx$$
Then it looks like the integration method of doing Gamma distribution, by u-substitution, let $u = \beta e^{-\gamma x}$, $du = -\beta \gamma e^{-\gamma x}dx = -\gamma u dx$, plug in the original formula, we get
$$\int_{-\infty}^\infty \alpha e^{\beta e^{-\gamma x}}dx = \int_0^\infty -\frac{\alpha}{\gamma u} e^{-u}du=-\frac{\alpha}{\gamma}\int_0^\infty u^{0-1} e^{-u}du=-\frac{\alpha}{\gamma} \cdot \Gamma(0)=-\frac{\alpha}{\gamma}$$
Here I assumed $\Gamma(0) = 1$, and you should double check whether I'm right. But generally it is the idea.  
