There is actually no particular reason why exponential growth must be
expressed as a power of $e$.
In the example of radioactive decay, for example, we often speak of
the half-life of a radioactive isotope, which is the amount of time
that it takes for half of the atoms in a sample to decay.
Thus if the half-life of an isotope is $H$ years, the portion of the sample
that will remain after $t$ years is $2^{t/H}$ —a power of $2$,
not expressed as a power of $e$.
Likewise, in finance there are things such as the "rule of $70$"
for estimating the amount of time it will take for an amount of money
to double at compound interest. If the result of the rule of $70$ is
$D$ years, and you leave $P$ dollars in an account at the same rate
of compound interest for $t$ years, the rule of $70$ says
you have $P \times 2^{t/D}$ dollars in the account at the end.
Again, a power of $2$, no $e$ in sight.
Of course you can easily convert any exponential growth or decay law
into a power of $e$ law: just introduce a constant factor in the exponent,
or change the constant that is already there.
For example, $2^x = e^{x \ln 2}$.
The particularly nice thing about $e$ that makes people want to use powers
of $e$ rather than powers of $2$ or $10$ in so many places is not
its exponential growth for large exponents, but the particularly
nice way it behaves for very small exponents.
In many places in mathematics and science where the exponent of something
is a variable, that exponent is not necessarily a whole number.
So yes, $2^3 = 2 \times 2 \times 2$ and $e^3 = e \times e \times e$,
but $2^{3/2} = 2 \sqrt 2$ (why?), $2^{1.4} = 2 \sqrt[5]{2^2}$,
and $2^{\sqrt2}$ … that's not so easy to write in other terms,
but it's a number between $2 \sqrt[5]{2^2}$ and $2 \sqrt 2$,
and it is well-defined as the limit of $2^x$ for real numbers $x$
as $x$ approaches $\sqrt2$.
By a small exponent I mean something like $0.001$, or better still
$0.000001$, or even $10^{-17}$.
The exponential function of any positive real number $R$ (where $R$ could
be $2$, $10$, or $289$, not just $e$) has the nice feature that when
$x$ is small, you can write
$$R^x \approx 1 + kx,$$
where $k$ is a constant that depends on $R$ but not on $x$,
and this approximation is very good if $x$ is small enough.
The special thing about $e$ is that for small $x$,
$$e^x \approx 1 + x,$$
that is, the constant $k$ is not needed. (Or you could say $k=1$.)
This special feature of $e$ is intimately related to all kinds of other
special properties it has, such as the fact that $\frac{d}{dx}e^x = e^x$,
or that $\int \frac1x \;dx = \log_e x$, or the way $e$
helps in financial calculations.
This nice property is also a reason why $e^{ix}$, where $x$ is a real number,
can be viewed as a rotation around the center of the complex plane.
For very small $x$, $ix$ is also small in the way that makes this a
good approximation:
$$e^{ix} = 1 + ix.$$
This is a number close to $1$, but a little bit "off to one side"
from the real-number axis. And the thing is that if we take any
point on the unit circle in the complex plane—the numbers of the
form $\cos\theta + i \sin\theta$—and multiply by $1 + ix$,
for small $x$, the result is approximately
$\cos(\theta + x) + i \sin(\theta + x)$, and if we multiply by $e^{ix}$,
that is the exact result:
$$e^{ix}(\cos\theta + i \sin\theta) = \cos(\theta + x) + i \sin(\theta + x).$$
That is, multiplication by $e^{ix}$ moves a point $x$ radians around
the unit circle.
The unit circle is $2\pi$ radians from $1$ all the way around and back to $1$
again, so $e^{i2\pi} = 1$ Halfway around the circle is $\pi$ radians,
and the point opposite $1$ on the circle is $-1$, so $e^{i\pi} = -1$.
That's it. Multiplication by $e^{ix}$ rotates a complex number $x$ radians
around the center of the plane, and the number $e^{ix}$ itself is the
point on the complex plane that $1$ gets to if you turn it $x$ radians
around the origin.
