Let's assume there is some non-standard model of the reals containing a number $N$ that is larger than any real number.
Suppose $\exists N\in {^*}\mathbb{R} ( \forall r\in\mathbb{R}: r<N).$
Now I know I can find even larger unlimited hyper-reals $2N,3N...$ which are smaller than $N^2,N^3...$ which are still hyper-reals because of closure under multiplication.
However, for ordinal numbers I have heard that matters become tricky around the $\varepsilon = \omega^{\omega^{\omega^{...^{...}}}}$ numbers.
So given my unlimited $N$ I think I should be able to reach $N^N, N\uparrow\uparrow N$ and so on. As far as I can work out, Knuth's arrow notation is defined recursively in $\mathcal{L}_\mathbb{R}$, so I think $N\uparrow^n N$ is well defined for finite $n$. I am less certain if it will transfer and give me $N\uparrow^N N$.
Something like the process used to compute Rayo's number on the other hand, seems outside of the bounds of $\mathcal{L}_\mathbb{R}$, so I would be very surprised if it could be generalised to make large hyperreals.
Moreover, is there some function (maybe $Tree(n)?)$, that is no longer defined for hypernatural inputs? And is there a sense in which the result of such a function could be "too big to be a hyper-real"?
I ask this last question because it seems possible to me that the answer could depend on which particular hyper-real field is being studied.
I think the most convincing way of answering this question is in an axiomatic approach to nonstandard analysis, such as Nelson's IST. Here infinitesimals (as well as unlimited numbers) are found within the "ordinary" real number line $\mathbb R$. They are detected by a new one-place predicate "standard". Thus, a positive infinitesimal is a real number smaller than every positive standard real number. An $N$ such as you are describing is a nonstandard natural number (bigger than every standard natural number). IST is a conservative extension of ZFC.
From this point of view, it is evident that all the usual constructions you mentioned carry over, and you can construct as big a number compared to $N$ as you wish, by all the usual techniques.
Provided such constructions do not use the axiom of choice, they can also be carried out in the framework SPOT which is conservative over ZF.