Applying power set finite times

Question

Is every infinite set $A$ smaller than a set of the form $\mathcal P (\mathcal P(\dots \mathcal P(\mathbb N)))$?

Answer 1

Consider the set $\bigcup_{k\in \mathbb N} \mathcal P^k(\mathbb N)$. This is larger than any $\mathcal P^k(\mathbb N)$.

Answer 2

Let $A$ be the following set: $\{X\mid\exists n\in\Bbb N:X\in\mathcal P^n(\Bbb N)\}$. Clearly every finite application of the power set operation is a subset of $A$, so I cannot have cardinality strictly smaller than any such set.

So why is $A$ a set at all? For this you need the axiom of replacement, and then we can argue that $F(n)=\mathcal P^n(\Bbb N)$ defines a function on $\Bbb N$, and therefore $\{F(n)\mid n\in\Bbb N\}=\operatorname{rng}(F)$ is a set, then $A$ is a set again by the axiom of union.