  In this survey, we present five different proofs for the transcendence of
Kempner's number, defined by the infinite series $\sum_{n=0}^{\infty}
\frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas
and methods that are used for proving deeper results. We outline proofs for
some of these results and also point out references where the reader can find
all the details.
