We all know about pi, and how this mysterious constant has a natural analogue – it’s the ratio of the circumference of a circle to its diameter.
But now about e? What’s natural about that? There’s a great explanation here at the site Better Explained
e is the base rate of growth shared by all continually growing processes.
Yes, you say, but don’t things grow at different rates? Sure, but rate and time are just modifiers to this basic growth rate structure. For a better explanation than I can give here without plagarism, see the link above.
(thanks to Marginal Revolution for the link).