Let $X_0,\dots,X_n,\dots$ be a sequence of independent variables and let each $X_k$ be distributed according to an exponential distribution of parameter $\lambda a^{-k}$ (with $a<1$):
$$X_k \sim \text{Exponential}(\lambda a^{-k}) \Leftrightarrow \mathbb{P}(X_k<x) = 1- \exp(\lambda a^{-k} x)$$
Now define:
$$Y = \sum_{k=0}^\infty X_k $$
It is straightforward to show that the variable $Y$ has finite expectation and finite variance:
$$ \mu_Y = \mathbb{E}(Y) = \frac{1}{\lambda}\frac{1}{1-a}, \qquad \sigma_Y^2 = \mathbb{E}(Y^2)-\mathbb{E}^2(Y) < \infty$$
Moreover, for small values of n it is possible to calculate in a simple way the distribution of $Y_N$
$$Y_N = \sum_{k=0}^N X_k$$
using the characteristic function formalism. But for N tending to infinity, it does not seem to express the distribution of $Y$ as closed form. I have simulated that $Y$ is distributed as follows a Fréchet distribution, which makes sense since it is an extreme value distribution, which appears in the Fisher–Tippett–Gnedenko theorem. Note that $\tilde{N}_t = \#\{i \ge 1: Y_N \le t \}$ is an explosive Cox Process.
How could it be formally demonstrated que $Y \sim \text{Fréchet}(\cdot,\cdot)$ ?
Additional comment: The characteristic function of $Y_N$ is given by:
$$\phi_{Y_n} = \mathbb{E}(e^{it(X_1+\dots+X_n}) = \prod_{i=k}^N \left(1-\frac{it}{\lambda a^{-k}}\right)^{-1}$$
for low $N$ is relatively easy to compute the Fourier transform of this function to obtain the probability density function of $Y_N$.