How many cardinals are there? I'm trying to do the following exercise:

EXERCISE 9(X): Is there a natural end to this process of forming new infinite cardinals? We recommend this exercise instead of counting sheep when you have trouble falling asleep.

(This is from W. Just and M. Weese, Discovering Modern Set Theory, vol.1, p.34.)
By this process they mean $|\mathbb N| < |\mathcal P(\mathbb N)| < |\mathcal P (\mathcal P (\mathbb N))| < \dots$. My first response to this was "Obviously there is no end to it." but then the exercise is supposed to be challenging ("X-rated") so this must be wrong and there is an end to it. But when exactly? How many cardinals are there? What would be a "natural end"? Thank you for your help!
 A: I don’t know what they have in mind, but the process as described goes only $\omega$ steps. To go further, you have take the union of what you already have and start over. That is, the process of repeatedly taking the power set will get you $\beth_0,\beth_1,\beth_2,\dots$ and hence $\beth_n$ for each $n\in\omega$, but it won’t get you beyond those. If you take the union of all those power sets, you get something whose cardinality is $\beth_\omega$, and you can start powering up again.
A: I think what this exercise invites you to contemplate is what "a natural end" could even mean here.
It is clear that the $|\mathcal P^n(\mathbb N)|$ sequence gives you a countable infinity of different infinite cardinals, but is that an "end"? And if it is, would it be a "natural" one? "Countably infinite" is, as you know by now, a rather poor and sickly kind of infinity. Could we have continuum many different cardinals, for example?
If you have learned about initial ordinals, you might consider what this concept implies for the order type of the collection of all cardinals with their inherent order.
A: I think it is important to consider the context in which this question was posed.
We first have Cantor's Theorem (Theorem 5, p.32) which says that $| A | < | \mathcal{P} (A) |$ for any set $A$.
But we also have Theorem 7 (p.34), which I will quote in its entirety:

Let $\mathcal{M}$ be a set of cardinals such that $\forall \mathfrak{m} \in \mathcal{M} \exists \mathfrak{n} \in \mathcal{M} ( \mathfrak{m} < \mathfrak{n} )$, and let $\mathcal{A}$ be a family of sets such that $\forall \mathfrak{m} \in \mathcal{M} \exists A \in \mathcal{A} ( |A| = \mathfrak{m} )$.  Then $| \bigcup \mathcal{A} |$ exceeds every cardinal in $\mathcal{M}$.

It is this last ingredient that allows this process to extend beyond the simple successor cases.  (And this is what everyone else has been saying.)
Start with $X_0 , X_1 = \mathcal{P}(X_0) , X_2 = \mathcal{P} (X_1) , \ldots , X_{n+1} = \mathcal{P} ( X_n ) , \ldots $ and then after $\omega$ steps take $X_\omega = \bigcup_{n < \omega} X_n$.  By Theorem $7$ we know that $| X_\omega | > | X_n |$ for all $n < \omega$.  We continue with $X_{\omega + 1} , X_{\omega_2} , \ldots , X_{\omega \cdot 2} , X_{\omega \cdot 3} , \ldots , X_{\omega \cdot \omega}$ etc.  We can even imagine this extending beyond the countable ordinals, arriving at $X_{\omega_1}$ which will satisfy $| X_{\omega_1} | > | X_\alpha |$ for all $\alpha < \omega_1$.  And this can continue still more.
A: Assume there is an end and $A$ is a set of maximal cardinality. Then you still have $|\mathcal P(A)|>|A|$, contradicton.
It is even worse: The "set" of cardinalities is not a set - because it could not possibly have a cardinality. (Though there is some longer story behind that).
A: This is a matter of philosophical approach to what it means "a natural end". After all, there is no natural end to the generation of natural numbers -- but certainly they end somewhere because we know that most ordinals are uncountable.
On the same note generation of cardinals does not end as long as you have ordinals to iterate through (either in the power set operation; or in the Hartogs number operation; or limits of these constructions). In this sense there is no natural end to the process of generating new cardinals.
However if one thinks of operations performed along the class of ordinals as operations which terminate when you "run out of ordinals", and if you can run out of natural numbers there is no reason you cannot run out of ordinals as well, in such case there is in fact a natural ending to the generation of new sets and therefore of new cardinals.
