Exponential nature of current growth/decay in inductors
This is quite the question.

Let me start by saying that the factor \$e\$ has been invented to make our life a bit easier. Any power can be written as another power with a different base as follows:

$$
a^b = \left(e^{\ln(a)}\right)^b = e^{\ln(a)\cdot b}
$$

The usual way how this relationship occurs is when a quantity changes linearly according to its own value in some way. Ie. if the quantity increases, then the rate-of-change of that quantity will change proportionally.

Your question could be considered invalid though, because there are a lot of circuits with inductors and capacitors that don't change exponentially. For example:







The current through the capacitor is constant, and the voltage changes linearly. There is no exponential involved.

An example of where it does seem to be the case would be:







(V1 is a step function)

The current through the capacitor depends linearly on its own voltage, because of the added resistor. The behavior of a capacitor is such that \$i = C\frac{dv}{dt}\$ so this means a derivative will ultimately be linearly dependent on its own value. This circuit will result in a solution with an exponential.

Rather than talking about flux, voltage, current, etc. you should consider that having a solution that is exponentially changing is something much more basic.