A number $x$ is called normal in base $b$ if every sequence of base $b$ digits $b_1b_2...b_n$ occurs with natural density $1/b^n$ in the decimal expansion of $x$.
There exist numbers normal in every base (called absolutely normal) and irrational numbers normal in no base (called absolutely non-normal), an example is given here.
Is it known whether there exist numbers that are normal in every base except one or numbers non-normal in every base except one?
The question can be stated rather easily but an answer probably will take a lot of effort, so thanks in advance, also for any reference to literature :).