Does $\sqrt{2}$ contain every digit in every base?
This popped into my mind and I have no idea how to begin attacking it.
Each digit has to occur only once.
Of course, $\sqrt{2}$ is probably normal in every base, but all I want is for each digit to occur at least once.
The same question for almost all reals.
We don't know whether $\sqrt2$ is a normal number, so we also don't know the answer to your question.
In fact, only very few numbers have been shown to be normal. As example is Chaitin's constant.