I guess compound interest could be considered indefinitely exponential, but you eventually reach a barrier in what's insured and it is a relatively small exponent. Also, is it still savings if you never spend it? I wonder what is the longest continuous account in banking that has never been touched.
Anyway, that is a tangent and a somewhat artificial scenario. Can you name a naturally occurring scenario? I would accept technology if you could show that it isn't going to level out like all other natural phenomenon.
Who says it needs to be exponential forever?
(Humans, fortunately, are starting to get better about this.)
Also, you want "logarithmic," not "logistic."
Don't think so - https://en.wikipedia.org/wiki/Logistic_function
The 'S' curve of logistic growth looks exponential for a while, which is why the question arises. By contrast, no one mistakes logarithmic growth for exponential growth for very long.