TITLE: A closed-form irrational number with few distinct digits
QUESTION [7 upvotes]: 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).

REPLY [5 votes]: Regarding hard: it is conjectured that each irrational element of the ring of periods is a normal number. So, they are excluded. This already excludes a lot. $e$ is also conjectured to be normal. Basically, since everything but a set of (Lebesgue) measure zero is normal, you should really have to resort to constructions (that are in some sense artificial) to avoid normal numbers. 
So, this excludes most notions of closed form.