"Every bounded function that is holomorphic on $A$ is is constant."
For which $A\subseteq\mathbb{C}$ is this true?
Are there well-known examples of unbounded sets $A\subseteq\mathbb{C}$ on which there are non-constant bounded holomorphic functions?
Later edit: My striking through the second question was meant only to de-emphasize it. Feel free to post further on it if you wish. Some of the examples posted in response to it were already well known to me; I'd have thought of them if my attention had been on the second question rather than the first.
I'm envisioning a couple of possibilities: (1) Various other sorts of sets $A$ will be mentioned in answers; and (2) An answer will say that some nice theorem says this is true of a set $A$ if and only if whatever, where "whatever" is something non-trivially different from a tautologous "if and only if every bounded holomorphic function on $A$ is constant", and maybe "whatever" is somehow elegant or at least simple.