What's $(\omega\cdot2)\times(\omega\cdot 2)$?
Then $(\omega\cdot2)\times(\omega\cdot 2)\times(\omega\cdot 2)$?
Then the limit of this sequence?
I think it's $(\omega\cdot 2)^\omega$ but have very limited experience with ordinals. Is that even an ordinal, and is there a better way to write it?
This part of a larger problem that I'm breaking down. If there were any hints or comments on that, I could maybe have clarified there.
To answer the question:
If we take the cartesian product of $\omega$ many sets $(\omega\cdot 2)$, we get the set of all sequences of the form $\langle \alpha_n\mid n\in\omega\rangle$ where every $\alpha_n\in\omega\cdot 2$. Another way to see this is as the set $^\omega(\omega\cdot 2)$ of all functions $\omega\to\omega\cdot 2$.
If we use ordinal multiplication, then $\omega\cdot 2\cdot\omega\cdot 2=\omega\cdot\omega\cdot 2=\omega^2\cdot 2$. In general for any $n\in\omega$ you will get $\omega^n\cdot 2$ after $n$ multiplications.
The limit $(\omega\cdot 2)^\omega$ then is equal to $\sup_{n\in\omega}\{\omega^n\cdot 2\}$. Note that this is equal to $\omega^\omega$, since it is indeed an upper bound, and any ordinal less than $\omega^\omega$ will be of the form $\omega^k\cdot \beta+\alpha$ for some $\alpha<\omega^k$, $\beta<\omega$ and $k\in\omega$, and cannot be an upper bound.