Suppose you are given a number $0 < c < 1$, and an i.i.d. sequence of random variables $x_i$, such that $P \left[ 0 \le x_i \le 1  c \right] = 1$. The series $\sum_{k=0}^\infty x_k c^k$ converges a.s. to some random variable $\xi$ with $P \left[ 0 \le \xi \le 1 \right] = 1$. I'd like to find examples where we can identify the distirbution of $\xi$ explicitly (it is easy to compute all moments recursively, but that's not what I am talking about). One example is given by $c=\frac{1}{N}$, and $x_i$ having a discrete distribution on $ 0, 1, 2, \ldots , N1 $ ($N$ is a natural number): the limit is uniform on $[0,1]$. Any other case?

One thing that naturally comes to mind is the Cantor staircase distribution (where $c=1/3$ and $x_i$ takes values $0$ and $2/3$ with probability $1/2$) and its immediate generalizations. Other than that I don't think much is known explicitly. You also certainly can write down the characteristic function as an infinite product. You also can view the distribution as an invariant distribution for process $Z_{n+1}=cZ_{n}+x_{n+1}$. 


One particular (relatively) well studied case is that of Bernoulli convolution, where the RV are supported on just 2 points. Take a look here to get an idea about what is known. 


The Fabius function is the cumulative distribution function of a sum of exponentially scaled uniform distributions $$\sum_{i=1}^\infty \frac{U_i(0,1)} {2^i}.$$ This is an example of a somewhat natural smooth function which is not analytic at any point of $[0,1]$. 

