I have two in one question:
1) Let $\{p_n\}_{n\in \mathbb{N}}$ be sequence of all prime numbers. Is number $\displaystyle\alpha = \sum_{n=1}^{\infty} 10^{-p_n}$ transcendental number?
2) Let $\{F_n\}_{n \in \mathbb{N}}$ be Fibonacci sequence with initial values $F_1=1$ and $F_2=2$. Is number $\displaystyle\beta = \sum_{n=1}^{\infty} 10^{-F_n}$ transcendental number?
If anyone thinks of something better to retag it with, be my guest.
Your second number has been proved transcendental. A somewhat more general result about numbers of the form $\sum 10^{-\lfloor \beta^k\rfloor}$ is mentioned in the Wikipedia list of numbers that have been proved transcendental.
Added: It has been pointed out that the Wikipedia reference is, to put it mildly, thin. To chase down information about Sturmian sequences, one can for example look at this article and its references. I believe that the original result is due to Mahler.