Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ preserving order.
Is this correct?
If that is the case. are there any relevant conclusions from this? What systems prove this for what ordinal $\alpha$?
Leaving aside for a moment the issues with class functions, here's an easy definition of such an $f$ for any $\alpha$:
$f(0)=0$,
$f(1+\beta)=\alpha+\beta$ for all $\beta\in ON$.
This $f$ is extremely easy to formally define, as long as we can make sense of ordinal addition: it is defined by the formula $$f(\alpha)=\gamma\iff (\alpha=0\wedge \gamma=0)\vee \exists \beta<\alpha(\alpha=1+\beta\wedge \gamma=\alpha+\beta).$$ Any theory capable of proving even very basic facts about ordinals shows that this formula defines a total function; and if we're working in a class theory, then a very small amount of class comprehension proves that this function exists.