The 'popular mathematics' literature (think Martin Gardner, William Dunham, Hofstadter, and the like) abounds with material on the mathematics of infinite cardinals, starting - and quite often ending - with Cantor's uncountability proof(s). You can't turn round in that section of a bookshop without bumping into a diagonalisation argument, or Hilbert's quote about paradise.
But what I've never seen, and what I would like to see, is similarly-pitched material about infinite ordinals. At the risk of being hand-wavey, if you can imagine an article about cardinals starting approximately
Let's define cardinality by {explanation}, and explore what happens if we extend the idea to infinite sets; give the name $\aleph_0$ to the cardinality of $\mathbb N$, and off we go
then I'd like to see something starting
We know how to count $1, 2, 3$. Now let's say that there's something called $\omega$ that's {explanation}, and off we go
My specific motivation is to get some kind of intuition for what's going on in the proof of Goodstein's theorem, but of course all knowledge is its own reward...