In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an iteration of the other. The sequence of building such iterated operators can go further to define faster and faster hyperoperators. The first of them is tetration which is defined as iterated exponentiation. Let $m\uparrow n$ denote the tetration of $m$ and $n$ that is $\underbrace{m^{m^{m^{.^{.^{.}}}}}}_{n – times}$. This operator appears in several interesting occasions in logic, computations and combiantorics, for example see these Wikipedia articles on Graham's number, Ackermann's function, busy Beaver function and Chaitin's incompleteness theorem.
Now consider the infinitary case. In set theory addition, multiplication and exponentiation are defined for cardinal numbers.
Question 1. What about $\kappa\uparrow\lambda$? How should we define this?
Intuitively, we expect to define $\aleph_0\uparrow\aleph_0$ to be $\aleph_0^{\aleph_0^{\aleph_0^{.^{.^{.}}}}}.$
But this intuitive definition of tetration has some counter-intuitive properties, as then we expect to have $\aleph_0^{(\aleph_0\uparrow\aleph_0)}=\aleph_0\uparrow\aleph_0$ which is impossible by Cantor's theorem which says $\forall \kappa\geq\aleph_0\;\;\;\aleph_0^{\kappa}>\kappa$.
Note that for the cases of addition, multiplication and exponentiation, we have quite natural operations $f_+, f_\times$ and $f_e$ such that given cardinals $\kappa, \lambda$, we have $f_+(\kappa,\lambda)=\kappa+\lambda, f_\times(\kappa, \lambda)=\kappa\times \lambda$ and $f_e(\kappa,\lambda)=\kappa^\lambda.$
Question 2. Is there a natural operation $f_t$ defined so that for all natural numbers $m,n$ we have $f_t(m,n)= m\uparrow n$, and so that its definition is so natural that it also works for infinite cardinal numbers?
The next question is taken from Noah's answer, where an answer to it may help in defining the tetration for higher infinite.
Question 3. What is $m\uparrow n$ counting?
See also What combinatorial quantity the tetration of two natural numbers represents?. But note that the answers given in the above question are so that they are not suitable for treating infinite cardinals.
Best Answer
Let me begin with an observation:
[If you followed the combinatorics tag here and have forgotten what the rank of a set is, see this article for a refresher. Roughly, the rank of a set is the "depth" of the nesting of braces when it's written out in long form. For example, we have one set of rank $0$ (namely $\emptyset = \{\}$). There are two sets of rank $0$ or $1$ (namely $\{\}$ and $\{\{\}\}$), there are $4$ sets of rank at most $2$ (namely $\{\}$, $\{\{\}\}$, $\{\{\{\}\}\}$, and $\{\{\},\{\{\}\}\}$), there are $16$ of rank at most $3$, $65536$ of rank $\leq 4$, etc.]
This observation is easily proved by induction, and it gives us a natural answer to your Question 3 when we restrict ourselves to tetration with base 2.
What about the unrestricted version of Question 3?
For this, we want to stop viewing set membership as a binary relation (membership value of $0$ or $1$). Instead, let's allow $k$ possible membership values: a set can be a member of another with value $0$ (not a member), $k-1$ (fully a member), or anything in between.
For example, this view still gives us one set of rank $0$, namely the empty set $\{\}$, but now we have $k$ sets of rank $0$ or $1$, namely the sets containing only $\{\}$, with value between $0$ and $k-1$ (of course, the set containing $\{\}$ with value $0$ is identified with $\{\}$, so that we only get $k-1$ new sets of rank $1$, for a total of $k$ with rank $0$ or $1$). It is easy to check that we get $k^k$ "sets" of rank at most $2$, $k^{k^k}$ of rank at most $3$, and $k \!\uparrow\uparrow\! n$ sets of rank at most $n$.
So a plausible answer to your Question 3 is:
One of the nice things about this answer is that it generalizes easily to higher cardinals.
As you know, the notion of $B$-valued set membership, where $B$ is some set (usually a Boolean algebra or a partial order) is a common one in set theory. Sets with a $B$-valued notion of membership are often called $B$-names, and they are naturally equipped with a notion of rank. In light of the foregoing discussion on finite tetration, I propose the following as a fairly natural definition for transfinite tetration:
Notice that this definition coincides with the inductive definition in Noah's answer. One drawback to this definition (noticed already by Noah) is that the value of $\kappa \!\uparrow\uparrow\!\nu$ is determined by treating $\kappa$ as a cardinal while treating $\nu$ as an ordinal.
Right now, I can see two approaches to "fixing" this drawback: either live with it (it's not a bug, it's a feature!), or modify the definition to (something like) one of the following three possibilities (the third being my personal preference):