Let me add more examples:

**(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, $F$ can not be constant.

**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.