It is shown that there exists a Riemann surface on which every Dirichlet finite harmonic function is automatically bounded and yet the linear dimension of the linear space of Dirichlet finite harmonic functions on it is infinite.
"Surfaces carrying sufficiently many Dirichlet finite harmonic functions that are automatically bounded." J. Math. Soc. Japan 64 (1) 201 - 229, January, 2012. https://doi.org/10.2969/jmsj/06410201