After looking at this question for quite some time, I've asked a couple of other students, and they also couldn't seem to come up with an answer. This is from an old qualifying exam at our university.
Let $u$ be a harmonic function bounded on the set $\{z:0 < |z| < 1\}$. Can it always be defined at the point $ z= 0$ to become harmonic on the whole unit disk?
The standard argument with the logarithm doesn't work here, as it must be bounded on the punctured disk, and the logarithm blows up at $0$. Also, we can't use any kind of harmonic conjugate argument because our domain is not simply connected. Thus, I think it's probably true, but I haven't been able to come up with a proof.
Thanks in advance for any help!