Let me add more examples:
If we consider the global behavior of the power function, then we have for example:
(A) (Foreman-Woodin): $F$ can be such that $F(\alpha)>\alpha+\omega,$ all $\alpha$ (modulo a supercompact and infinitely many inaccessibles above it). Note that by a result of Patai, there is no $\beta>\omega$ such that $F(\alpha)=\alpha+\beta,$ all $\alpha$.
Remark. In the above model, $F$ is not definable from the ground model, but we can go to intermediate submodel in which $F$ is definable.
(B) (Cummings): $F$ can be such that $F(\alpha)=\alpha+1,$ all successor $\alpha,$ and $F(\alpha)=\alpha+2,$ all limit $\alpha$ (modulo a $\kappa+3$-strong cardinal $\kappa$. By work of Gitik-Mitchell, we need more than a $\kappa+2$-strong cardinal $\kappa$).
(C) (Merimovich): Let $2\leq n < \omega.$ Then $F$ can be taken to be $F(\alpha)=\alpha+n,$ all $\alpha$ (modulo a $\kappa+n+1$-strong cardinal $\kappa$. By work of Gitik-Mitchell, we need more than a $\kappa+n$-strong cardinal $\kappa$).
(D) (Firedman-G): We can have (B) or (C) just by adding a single real to a model satisfying $GCH$. More precisely, the final model can be of the form $V[R],$ where $V\models GCH$ and $R$ is a real.
If we consider the local behavior of the power function, then we can say more:
(E) (Gitik-Merimovich): Let $2\leq m <\omega,$ and let $\phi: \omega\to \omega$ be such that $\phi$ is increasing and $\phi(n)>n,$ for all $n$. Then we can have $F(n)=\phi(n)$ and $F(\omega)=\omega+m$ (modulo a $\kappa+m$-strong cardinal $\kappa$).
(F) (Gitik): We can have $F$ defined on $\omega_1$ such that both sets $\{ \alpha<\omega_1: F(\alpha)=\alpha+2\}$ and $\{ \alpha<\omega_1: F(\alpha)=\alpha+3\}$ are stationary in $\omega_1$ (modulo suitable large cardinals. Some similar results are also proved by Gitik-Merimovich).
If we avoid choice, then an Easton like theorem is valid for all cardinals:
Let $\theta(\kappa)=sup\{\nu:$ there exists a surjection $f: p(\kappa)\to \nu \}.$ It is easily seen that $\theta(\kappa)>\kappa^+$ is a cardinal and it is increasing. The next theorem shows that these are the only restrictions that $ZF$ imposes on $θ(κ)$:
(G) (Fernengel-Koepke, based on an earlier result of Gitik-Koepke) Let $M$ be a ground model of $ZFC + GCH +$Global Choice. In $M$, let $F$ be a function defined on the class of infinite cardinals such that
i. $F(κ)$ is a cardinal > $κ^+$;
ii. $κ < λ$ implies $F (κ)\leq F (λ)$.
Then there is an extension $N$ of $M$ which satisfies $ZF$, preserves cardinals and cofinalities, and such that $θ (κ) = F (κ)$ holds for all cardinals in $N$.