Motivating e

Here's the background: my honor's class has done sequences and series, some basic function stuff (domain and range, composition and inverses) and some descriptive statistics. During the last week, we've also handled exponent rules, equations and functions. My textbook and syllabus are pushing for me to introduce e as "the natural exponent" next week, but how do I, at this stage, motivate that it's "natural"?
Continuous compounding is not a nice fit right now, and we're not even touching calculus until maybe late spring.

What other options are there?

Edit: oh, and I found this, and it's fantastic and I got the whole e-book and it's making me consider compounding after all.

Edit: I decided compounding may work and made this worksheet that students get as homework (for later class discussion).
I'm also giving something very similar to my regular class (seniors) who are doing financial math. Given my previous less-than-perfect (crash and burn) experiences with doing investigations with this class, I'd really like to get this right. It'll be optional, as e is not in their syllabus. Even so - any suggestions will be highly appreciated.

As always, google docs kills equations, so download the documents for best effect.