An example of a set with order type $\omega^2$ is $\mathbb{N}\times\mathbb{N}$ with a lexicographic order.
An example of a set with order type $\omega^3$ is $\mathbb{N}\times\mathbb{N}\times\mathbb{N}$ with a lexicographic order.
I was attempting to construct a set of order type $\omega^\omega$ and my first thought was to the extend the above pattern and say that the answer is the set of all sequences of natural numbers under lexicographic order. However that is obviously wrong, since the set of all sequences of natural numbers is uncountable while $\omega^\omega$ is countable.
What then is an example of a set with order type $\omega^\omega$? I'm not looking for anything natural or organic, just any example that would allow me to see what $\omega^\omega$ looks like. An example that continues the above pattern would be particularly nice.
First thing first, $\omega^\omega$ can denote the cardinal exponentiation or the ordinal exponentiation, and one usually has to understand that from the context (when they mix it is often best to use ${}^\kappa\lambda$ for cardinal exponentiation, or just use $\aleph$ numbers to denote cardinals.)
The ordinal exponentiation $\omega^\omega$ is the least ordinal after $\omega^n$ for all $n$. So you're looking for something which you get when you take $\omega^n$ and let $n$ grow to infinity.
So if $\Bbb{\omega=N,\omega^2=N\times N,\omega^3=N\times N\times N,\omega^\mathit n=\underbrace{N\times\ldots\times N}_{\mathit n\text{ times}}}$, and so on, what sort of ordered set can you think of which is the limit of these?