I was thinking about a nontrivial elem. embedding $j:V\rightarrow V$ (with a critical point), and I came across this question:
Given $cp(j)=\lambda$, is $\{\lambda,j(\lambda),j(j(\lambda))...\}$ cofinal in $\mathrm{Ord}$ (i.e. has no supremum)?
This question generalizes to whether or not $\mathrm{Ord}$ has a cofinality, and further generalizes to whether or not partially ordered proper classes have subclasses which are sets that are cofinal in them.
The question is how do you define the cofinality of a class.
If you define it as a "definable class function", then of course the cofinality is going to be $\rm Ord$ itself. Any function applied to a set results a set again, so it is bounded in $\rm Ord$.
But if you allow non-definable classes, then it will depend on the model, of course. If the height of the model is some regular cardinal, then you're going to have a "regular $\rm Ord$". But it can be singular, and not even a cardinal. So in that case, it can be just so.
To your question, that would strongly depend on the situation and the model. There is hardly any universal answer. Since any elementary $j\colon V\to V$ is not even second-order definable.