Don't want to sound condescending, but I don't think that seeing this in layman's terms has any merit. All of mathematics, although often built out of some intuitive, natural notions, is about definition: describe an object and then see what follows from your description. The description is not, in general, something that's "out there" and can be deduced by common-sense.

(Infinity, in fact, can be defined as a number and can be defined as a non-number in different contexts. But infinity is only something when it's part of a strictly defined mathematical universe with specific axioms.)

Yes and no. Your point is true, but not entirely relevant, becuase infinite sums, infinite products, and infinite process can (sometimes) be shown to have definite and consistent meanings and interpretations.

The infinite tower of exponentiation is not (of itself) the problem.

Infinitely small values (infinitesimals) can be quite useful:

http://en.wikipedia.org/wiki/Non-standard_analysis

The example doesn't claim that infinity is a number