Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are :
0.24813612512481361251248136125124813612512481371251249137125124913712512491371361 24913713612491371361
At first sight, the number appears to be rational because apparent patterns appear showing periods. In fact, the continued fraction of c has very large convergents. I calculated the first 20 000 digits from c with PARI and found a convergent with amazing 5817 digits! The terms afterwards are totally normal. This leads to the conjecture that c is transcendental. Has anyone an idea how this can be proven ?
A similar situation is observed in champerov's constant 0.12345678910111213... I read in the internet that this number also has extreme convergents without having obvious periods. Does anyone know why the large convergents occur ?
This not a complete answer.
One thing that can be said is that this number is not a normal one, as $1$ appears more often that $2$, $2$ more often that $3$, etc.
Also, it is definitely not rational, as the probability of a digit to be equal to $1$ is $\log_{10} 2$, which is an irrational number.
Also, it can be expressed as $$ \alpha=\sum_{n=1}^\infty \frac{\left\lfloor 10^{n\log_{10}2-\lfloor n\log_{10}2\rfloor}\right\rfloor}{10^n}. $$