Variance of $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$ Here the $X_i$'s are i.i.d. and such that convergence in distribution for the infinite sum, is guaranteed. Probably the easiest case is when $X_i$ has a Bernouilli($p$) distribution, then $Z$ has a discrete Geometric($1-p$) distribution. I am not interested in that case, but in more complicated examples. 
Actually, I am trying to find what kind of distributions $Z$ could have. I am sure the choices are very limited. Those that work are called attractors.  I tried a Normal($0, 1$) for $X_i$ and $Z$ has a nice, smooth distribution, but not normal. I tried log-normal too for $X_i$, but in that case there is no convergence. An attractor must satisfy the functional equation $F_Z = F_{X_1(1+Z)}$.
Most surprisingly, I tried $X_i  \sim \sin( \pi Y_i)$ with $Y_i$ Normal($0,1$). The resulting distribution for $Z$ is pretty smooth (but not normal), and its variance, based on strong empirical evidence, seems to be exactly equal to $1$. Is this really the case?  
I am also interested in the conditions required to have convergence in distribution for the infinite sum. 
Update
The reason I failed to obtain convergence with a log-normal $X_i$ is because I picked one with $E(X_i)>1$. More on this to be published in Part II of this problem, here. 
 A: Final update on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here.
Surprisingly, there is a simple and general answer to this problem, despite the fact that all the terms in the infinite sum defining $Z$, are correlated. First, let us assume that $|E(X_i)| < 1$. This is required for convergence. Let us also assume that $E(X_i^2)<1$. This guarantees that the variance exists. 
We have the following formula for the $k$-th moment, for $k\geq 0$:
$$E(Z^k) = E[(X_i(1+Z))^k]=E(X_i^k)E[(1+Z)^k].$$
It can be re-written as
$$E(Z^k) =\frac{E(X_1^k)}{1-E(X_1^k)} \cdot\sum_{j=0}^{k-1} \frac{k!}{j!(k-j)}E(Z^j).$$
I suspect much simpler recurrence formulas can be found, for $E(Z^k)$. It follows immediately that $E(Z)=E(X_i)/(1-E(X_i))$. Moments of order 2, 3, and so on can be obtained iteratively. A little computation shows that
$$Var(Z) = \frac{Var(X_i)}{(1-E(X_i^2))(1-E(X_i))^2}.$$
I checked the formula when $X_i$ is Bernouilli($p$), and it is exactly correct. I also checked empirically when $X_i$ is Uniform$[0,1]$, and it looks correct: $Var(Z) = 0.506$ based on 20,000 simulated $Z$ deviates, while the true value (according to my formula) should be $\frac{1}{2}$.
Now let's look at $X_i = \sin(\pi Y_i)$ with $Y_i \sim$ Normal($0,1$). There is some simplification due to $E(X_i) = 0$ in this case: $Var(Z) =  E(X_i^2) / (1 - E(X_i^2))$. To prove that $Var(Z)=1$ amounts to proving that $E(X_i^2) = 1/2$, that is:
$$E(X_i^2) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} x^2 \sin^2(\pi x) e^{-x^2/2} dx = \frac{1}{2} .$$
I computed this integral using WolframAlpha, see here. The approximation is excellent, at least the first 7 digits are correct. However, the exact answer is not $1/2$, but instead
$$E(X_i^2) = \frac{1+(4\pi^2-1)\exp(-2\pi^2)}{2} = 0.500000051...$$
Now, to answer the most challenging question - what kind of distribution is an attractor in this framework - we need to look at the formula that gives the moments of $Z$. Clearly, they can be pretty arbitrary, meaning that the class of attractors is very rich. Of course, not all sequences of numbers represent the moments of a distribution. In order to correspond to an actual distribution, moments most satisfy some conditions, see here. A less challenging question is to find a non-trivial distribution that can not be an attractor, that is a distribution that can never be the distribution of the infinite sum $Z$, no matter what the $X_i$'s are.  This is the object of the next section.
Distributions that can not be attractors
The distribution for $Z$ is highly constrained. It must satisfy a number of conditions, and thus, few distributions are attractors (though far more than in the central limit theorem framework, where by far the normal distribution is the main attractor, and the only one with a finite variance.) I'll give just one example here, for $Z$ distributions whose support domain is the set of all natural numbers.
Let us consider a very general discrete distribution for $X_i$, with $P(X_i = k) = p_k, k = 0, 1, 2$ and so on. In this case, $Z$'s distribution must also be discrete on the same support domain. This case covers all possible discrete distributions for $Z$, with support domain being the set of natural numbers. Let's use the notation $P(Z=k) = q_k$. Then we have:


*

*$P(Z=0) = p_0 = q_0 = P(X_1 =0)$,

*$P(Z=1) = p_1 p_0 = q_1 = P(X_1 = 1, X_2 =0)$,

*$P(Z=2) = (p_1^2 + p_2)p_0 = q_2 = P(X_1 = X_2 =1, X_3 =0)+P(X_1 = 2, X_2 =0)$,

*$P(Z=3) = (p_1^3 + 2 p_1 p_2 + p_3)p_0 = q_3$,

*$P(Z = 4) = (p_1^4  + 3 p_1^2 p_2  + 2 p_1 p_3+ p_2^2   + p_4 ) p_0 = q_4$.


We don't even need to use the third, fourth or firth  equation. Let's focus on the two first ones. The second one implies that $p_1 = q_1 / p_0 = q_1 / q_0$. Thus we must have $q_1 \leq q_0$ for $Z$ to be an attractor. In short any discrete distribution with $P(Z= 0) < P(Z = 1)$ is not an attractor. The geometric distribution is actually an attractor, the most obvious one, and possibly the only one with a simple representation. 
Another interesting question is the following: can two different $X_i$ distributions lead to the same attractor? In the case of the central limit theorem, this is true: whether you average exponential, Poisson, Bernoulli or uniform variables, you end up with a Gaussian variable - in this case the universal attractor; exceptions are few (the Lorenz distribution being one of them). The following section provides an answer for a specific attractor.
If $Z$ is the geometric attractor, then $X_i$ must be Bernouilli
Using the same notation as in the previous section, if $Z$ is geometric, then $P(Z = k) = q_k = q_0 (1-q_0)^k$. The equation $p_1 p_0 = q_1 = q_0(1-q_0)$ combined with $p_0 = q_0$ yields $p_1 = 1-q_0$. As a result, $p_0 + p_1 = q_0 + (1-q_0) =1$. Thus if $k> 1$ then $P(X_i = k) = p_k = 0$. This corresponds to a Bernouilli distribution for $X_i$.
Interestingly, the Lorenz attractor in the central limit theorem framework can only be attained if the $X_i$'s themselves have a Lorenz distribution. 
Connection with the Fixed-Point theorem for distributions
Consider $Z_k = X_k + X_{k} X_{k+1} + X_{k}  X_{k+1} X_{k+2}+ \cdots$. We have $Z_k = X_k (1+ Z_{k+1})$ .  As $k\rightarrow \infty, Z_k \rightarrow Z$. The convergence is in distribution. So at the limit, $Z \sim X_i(1+Z)$, that is, the distributions on both sides are identical. Also, $X_k$ is independent of $Z_{k+1}$. In other words, $Z$ (specifically, its distribution) is a fixed-point of the backward stochastic recurrence $Z_k = X_k (1+ Z_{k+1})$. Solving for $Z$ amounts to solving a stochastic integral equation.
