Let $\kappa$ be an infinite cardinal. Can I reach every intermediate cardinal $\mu$ with $\kappa \le \mu \le 2^\kappa$ as some power $\kappa^\lambda$?
If not, is there another construction that allows me to reach every cardinal between $\kappa$ and $2^\kappa$?
The power functions are a total wild card, as shown by Cohen, Solovay, Easton and others.
The only two things we can say about it, is that it is monotonous and increasing in cofinality.
More specifically, suppose that you start with a model of $\sf GCH$, add $\aleph_2$ Cohen reals, and you have a model where $2^{\aleph_0}=\aleph_2$ but $\aleph_1\neq\aleph_0^{\aleph_0}$ and of course that $\aleph_1\neq\aleph_0^n$ for any finite $n$.
And of course you can replace $\aleph_2$ here by any cardinal. Add "many" Cohen reals, and then every cardinal in the interval from $\aleph_0$ to $2^{\aleph_0}$ cannot be expressed like that.
It might be worth pointing out that $\sf GCH$ implies, on the other hand, that this is true, simply because $\mu=\kappa$ or $\mu=2^\kappa$ in that case. I suspect that you can prove that this assertion is in fact equivalent to $\sf GCH$.