My first idea was to find the largest $\kappa$ such that $2^{\aleph_0} = \aleph_{\kappa}$ is consistent with ZF but this set is unbounded ($\kappa$ can be any finite number) and $2^{\aleph_0} < \aleph_{\omega}$. Which brings up the question, how much fundamental difference are there between CH and $2^{\aleph_0} = \aleph_{118}$ for example?