I'm looking for an irrational number which does not have all $b$ distinct digits in its base-$b$ representation and which can be expressed in "closed form" for some reasonable definition thereof.
Acceptable would be +, -, *, /, ^, sin, $\pi$, $e$, and anything else 'reasonable'. Unacceptable would be algorithms, unbound $\sum$ or $\prod$, "write in base 2 and interpret in base $b$", etc.
The problem may be hard, in which case I would be willing to accept an answer explaining that this is so (ideally with a reference).