Simple (even toy) examples for uses of Ordinals?

You might find something useful in this post by Tim Gowers: http://www.dpmms.cam.ac.uk/~wtg10/ordinals.html. Especially his first example, with (countable) ordinals introduced as a convenient notation for indexing an increasing sequence of bounded increasing sequences (and so on in many levels perhaps), was quite illuminating for me.

That is, if $a_n \nearrow a$, and $a < b_n \nearrow b$, and $b < c_n \nearrow c$, etc., we will have the notational problem of running out of letters after a while. But we can instead write $a_{\omega}$ instead of $a$, and $a_{\omega+n}$ instead of $b_n$, and $a_{2\omega}$ instead of $b$, and $a_{2\omega+n}$ instead of $c_n$, etc., and thus index all the numbers using a single symbol $a$ with ordinals attached as subscripts. Even countably many sequences will not be a problem, since then we just denote the limit of the sequence $(a_{n\omega})_{n=1}^{\infty}$ by $a_{\omega^2}$. And so on...

I have only just found this question, but here is a "practical" application.

Suppose I say something will be ready within 7 days. That is represented by number 7. Each following day I say a number smaller than the day before, and when I say zero, that something is ready. On the second day I may say 6, or 3, or even 0.

How about if I say: "I'll tell you tomorrow how long it will take." or "Within 3 days I'll tell you how long it will take." - can that be represented by a generalized form of a number? Yes, it can: ordinal numbers.
First example: today I say $\omega$. That means tomorrow I will have to give you a finite number (let's call it $n$) because only finite numbers are smaller than $\omega$. Therefore you'll know tomorrow that it will be ready within $n$ days
Second example: today I say $\omega+2$. (No, it's not $\omega+3$.)

We can go further: within 7 days I will tell you when I will know how long it will take ($\omega\cdot2+6$) or even tomorrow I'll know when I'll know when I'll know when I'll know when it's ready ($\omega\cdot4$).
If we call the last one taking 4 stages, then imagine this: tomorrow I'll tell you how many stages it will take. That's $\omega^\omega$.

So ordinal numbers can express information about unknown waiting time that is definitely finite. Can everyone see why it is so? Or why the following is true? Start the sequence with any ordinal number and follow each number by a smaller one; it will eventually come to zero (in finite steps).