Whenever $\kappa$ is an infinite cardinal number, write $\mathrm{cl}_\kappa$ for the unique function $\mathrm{Card} \rightarrow \mathrm{Card}$ given by $\mathrm{cl}_\kappa(\nu) = \nu^\kappa.$ It follows that, whenever $\kappa$ is an infinite cardinal, we have that $\mathrm{cl}_\kappa$ is a closure operator (i.e. it is monotonic, and idempotent, and satisfies $\mathrm{cl}_\kappa(\nu) \geq \nu$). N.B. We need $\nu$ to be infinite for idempotency.
General Question. What are the fixed points of $\mathrm{cl}_\kappa$ as a function of $\kappa$?
Now the following is clear. Let $\kappa$ denote an infinite cardinal number. Then clearly, $0$ and $1$ are fixed points of $\mathrm{cl}_\kappa$. So too is $2^\kappa$; more generally, $2^\gamma$, whenever $\gamma$ is a cardinal greater than or equal to $\kappa$. I was thinking that maybe these are the only fixed points.
Specific Question. Are the fixed points of $\mathrm{cl}_\kappa$ precisely $0,1,$ and the cardinals that can be expressed in the form $2^\gamma$ for some $\gamma \geq \kappa$?
Beyond ZFC, feel free to assume that the continuum function $2 \mapsto 2^\kappa$ is injective, and/or that $2^\kappa$ is weakly inaccessible for every infinite cardinal $\kappa$. (I am not so interested in the implications of the continuum hypothesis or its strengthenings).
To the specific question, the answer is consistently no. For example consider the following case with $\kappa=\aleph_0$:
Suppose that $2^{\aleph_0}=\aleph_1,2^{\aleph_1}=2^{\aleph_2}=\aleph_3$. Now we have $\aleph_2^{\aleph_0}=\aleph_2\cdot2^{\aleph_0}=\aleph_2$, despite the fact that $\aleph_2\neq2^{\aleph_\alpha}$ for any $\alpha$. You can have that with $\sf ICF$ by having $2^{\aleph_n}=\aleph_{n+2}$ for $n>0$ and $2^{\aleph_0}=\aleph_1$ (and $\sf GCH$ above $\aleph_\omega$).
More generally there are some rules that would answer some part of your question in general, but I don't know of any nice characterization of such fixed points. There might be some more convoluted rules and division to various cases and so on. I can recommend the first chapter of the following book for references: