Is the following number transcendental? $$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)
This number is called the Copeland–Erdős constant, and is known to be irrational and normal. I believe its transcendence or otherwise is an open problem. This source claims that it has been proved to be transcendental, but the paper they refer to is the one in which it was proved to be normal and so I think the source is mistaken.
For now, the knowledge that it is almost surely transcendental will have to suffice!