When I hear people talking about something growing “exponentially” I think of the line from the Princess Bride where Inigo Montoya says

You keep using that word. I do not think it means what you think it means.

When most people say “exponential” they mean “really fast.” But that’s not what exponential growth means at all. Many things really *do* grow exponentially (for a while), but not everything that is “really fast” is exponential. Exponential growth can be amazingly fast, but it can also be excruciatingly slow. See Coping with exponential growth.

I like your note that nothing grows exponentially forever. In grad school I took a class in branching processes and we proved it. The professor noted that this is why pyramid schemes (Ponzi schemes) are doomed to fail.

More interesting was our analysis of chain mail propogation. The one we looked at was a common one which claims to have been around the world several times and to bring good fortune if reproduced and bad fortune if not. These claims were backed up by specific examples. IIRC this letter has been in circulation for decades in essentially unchanged form.

It can be shown that for this to be the case under any reasonable assumptions regarding the distribution of number of offspring letters, people must have been ressurecting the chain letter from time to time. That is to say, there must have been people who for whatever reason started mailing out the letter spontaneously, and that this must have been happening throughout the period that the letter was circulating. Otherwise the letter would have either died out or blown up to enormous circulation relatively quickly.

What I find more interesting is that so many things actually

dogrow exponentially for a while, sometimes for a long while.A simple biological example is population growth that is proportional to current population size. If there are no resource limits, the result is exponential growth. But in real life there are always resource limits, as well as all kinds of other phenomena like territoriality and predation. Most of the time what we call exponential growth is just *approximately* exponential growth. Where this really matters in when extrapolation is used to make predictions. Subtle deviations from exponential growth can lead to wildly inaccurate predictions.