You can find encodings in the natural numbers of certain collections of subsets of $\mathbb{N}$. For example, as was pointed out by Henning Makholm, there is a very nice encoding of the collection of finite subsets of $\mathbb{N}$.
More generally, let $A$ be a recursively enumerable subset of $\mathbb{N}$. Let $T_A$ be a Turing machine that, on input $n$, halts if $n\in A$, and does not halt otherwise. Then we can encode $A$ by using the usual index of the machine $T_A$.
This encoding is not fully satisfactory. There are infinitely many Turing machines that halt on input $n$ iff $n\in A$. Thus every recursively enumerable set has infinitely many encodings. Moreover, there is no algorithm for telling, given two natural numbers, whether they encode the same set.
For more modest collections than the collection of recursively enumerable sets, there are far more satisfactory encodings. As a small and not very interesting example, consider the collection of subsets of $\mathbb{N}$ that are either finite or cofinite (their complement is finite). A small modification of the encoding of finite subsets will take care of the finites and cofinites. Basically, just use the even natural numbers for the finites. If you have used $2k$ to encode a finite, use $2k+1$ to encode its complement.
In addition, the elements of any countably infinite subset of the power set of $\mathbb{N}$ can, by the definition of countably infinite, be encoded using the natural numbers. However, by cardinality considerations, we cannot encode all the subsets of $\mathbb{N}$ using elements of $\mathbb{N}$.
$2^\omega$ is exponentiation of ordinals, while $2^{\aleph_0}$ is exponentiation of cardinals (and that does give a larger cardinal than $\aleph_0$ corresponding to the power set of $\aleph_0$, that is what Cantor's argument is.)
$2^\omega$ is defined as the limit of the finite ordinals $2^n$ (because $\omega$ is the limit of the finite ordinals), and so is just $\omega$ again. Note that ordinal exponentiation works quite differently from the cardinal one, see here, e.g. or any good set theory text book.
Indeed $\omega_1$ is by definition the first uncountable (not in bijection with a subset of $\omega$) ordinal and hence (as cardinals are special ordinals) equal to $\aleph_1$. Operations on ordinals produce ordinals, and e.g. $\omega+1 \neq 1+\omega$ in ordinals, while in cardinals $\aleph_0 + 1 = 1 + \aleph_0 = \aleph_0$, etc. As sets $\aleph_0$ and $\omega$ are the same but the operations are different. One is used to measure sizes of sets, the other "measures" well-orders on sets. Beware of the differences!
Best Answer
Yes!
First and foremost, although the power set operation gives a larger cardinality, it need not give the next larger cardinality (I'm assuming that the cardinals are well-ordered here, which follows from ZFC). This is the heart of the Continuum Hypothesis: the reals are essentially the power set of the naturals, are there sets of reals of intermediate cardinality?
Second of all, we have the notion of limit cardinals. The following is a chain of sets of increasing cardinality:
$$\aleph_0, \mathcal{P}(\aleph_0), \mathcal{P}(\mathcal{P}(\aleph_0)), \dots$$
But what's the cardinality of their union? It's bigger than all of them, but it's not equinumerous with the power set of anything smaller than it. (Can you see why?)
The remaining question is, can we climb up past all the cardinals via repeated applications of power set and union? This leads to the notion of inaccessible cardinals. These are cardinals $\kappa$ which are larger than the power set of anything smaller cardinal $\lambda < \kappa$, and larger than any union of less than $\kappa$ many sets each of which has cardinality less than $\kappa$. That is:
The existence of such inaccessible cardinals is not provable from ZFC (unless ZFC is inconsistent). In fact, the hypothesis "there exists inaccessible cardinals" is so strong that ZFC + "there exist inaccessibles" implies the consistency of ZFC! This is of course strictly stronger, because ZFC alone cannot prove its own consistency, a la Godel. Inaccessible cardinals are one type of the famed large cardinals.