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).