Consider the number with binary or decimal expansion
that is, the $n$'th entry is $1$ iff $n$ is prime and zero else. This number is clearly irrational. Is it known whether it is transcendental?
Here, since I was "nominated" to answer, is what little I know.
The binary version is called the "prime constant" on some internet sites, but I am not aware of any substantial work on this number.
I initially found it by googling
Both binary and decimal versions are irrational because the expansions do not repeat, but I can offer no useful comment on the question of transcendentality. As Dan Brumleve notes, a Wikipedia article claims that the binary version is suspected transcendental. As Gerry Myerson notes, "every number not known to be algebraic is suspected transcendental."