The GCH axiom basically says that for all infinite cardinal numbers $\kappa$, the number of cardinals lying strictly between $\kappa$ and $2^\kappa$ is as small as possible. Namely, there are none.
Is there an axiom which claims the opposite, in other words that the number of cardinal numbers lying strictly between $\kappa$ and $2^\kappa$ is as large (in some sense) as possible?
Edit. For example - and I don't know if this is a silly suggestions, I know very little set theory - is the following axiom for infinite cardinals $\kappa$ consistent with ZFC? And if so, is it interesting? $$|\{\mbox{cardinals } \nu \mid \kappa<\nu<2^\kappa\}|=2^\kappa$$