The only numbers that have been proven to be normal are artificially constructed ones such as Champernowne's Constant (and its variants), the Copeland-Erdos constant. No specific constant has been proven to be absolutely normal (in all bases).
It has been conjectured that all irrational algebraic numbers are normal, but in base 10? Does the conjecture state that they are normal in all integer bases (absolutely normal)?
How would one actually go about proving this?