How can $e$ be used for practical mathematics? This is for a presentation on (among other numbers) $e$, aimed at people between the ages of 10 and 15.
To clarify what I want:
$e$ is not useful. It just happens to be the value at $x = 1$ of the exponential function $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
which is incredibly useful because it is its own derivative. This fundamental property helps us solve differential equations, which are a very powerful language for understanding the universe.
The boring example is modeling something like the growth of bacteria. (As Raymond Manzoni's comment says, a more interesting example involving exponential growth is carbon dating.) A much more interesting example (to me, anyway) occurs if you allow $x$ to be something more general than a real number: allow it to be complex and you get Euler's formula $$e^{ix} = \cos x + i \sin x$$
which tells you that the exponential can represent rotations in the plane. It also tells you that $\cos x$ and $\sin x$ can be used to describe simple harmonic motion, a basic and fundamental example of a differential equation describing real-world phenomena (such as circuits) that we know how to solve exactly thanks to the exponential function.
To be more specific, the current $I(t)$ in an LC circuit satisfies the differential equation
$$\frac{d^2 I}{dt^2} + \frac{1}{LC} I = 0$$
where $L$ is the inductance and $C$ is the capacitance. The complex exponential can be used to deduce that the solutions to this differential equation look like
$$I(t) = A \cos \omega t + B \sin \omega t$$
where $\omega^2 = \frac{1}{LC}$. This is a very precise description: the current is periodic and we can explicitly compute its period. Add in a resistor and you get a similar differential equation whose solutions can now be damped, and again we can explicitly compute how quickly the current is damped.
Exponentials are also closely related to the theory of the Fourier transform (which also helps us solve differential equations and much more besides), the normal distribution (which we use to understand statistics), approximating factorials (which we often need to do in computer science)...
$$\frac{1}{e}=\lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n$$
So, it appears in a large amount of games, for example, consider a game with $n$ players, where each player has $\frac{1}{n}$ (independent) chances of win. When $n$ growth, the probability that there is no winner is (very quickly) close to $\frac{1}{e}$.
It stays true, even for some cases where it is not independent. Play the game with $n$ balls (with numbers from $1$ to $n$, one on each ball), and $n$ players (each player has a different number). Then each player draws a ball in the bag (he can't see which ball he draws, and he keeps it). Once again, the chance of no one draws his own number is close to $\frac{1}{e}$ (for enough large $n$, a class of 15 children is enough)...
If, in such experiment, you make a redraw until there is no winner, the average number of draws needed is $e$.
If complex numbers are forbidden (removing much of the fun for the people here...) some practical applications :
natural exponential decay of radioactive components, concept of half-life (suggestive subject for children!). This is the solution of the simplest differential equation : $x'=-\lambda x$ giving $x=C e^{-\lambda t}$ (many practical application in Natural sciences in the link) exponential decrease of gaz density with altitude and so on...
simple ecologic system : predator-prey model considering a system of two coupled linear O.D.E system ($x$ is the number of predator and $y$ the number of preys) :
$x'=(b-py)x$
$y'=(rx-d)y$
(well solutions may have $\ln$ too or more complicated (chaotic) things but...)
use logarithmic scale to explain exponentials growth :
explaining orders of magnitude $e^{\ln(10)x}=10^x$ up to the scale of the universe (there is a nice video on the internet with powers of $10$)
I'm going to be a contrarian here. I think that it's a mistake to introduce the function $e^x$ prior to calculus. You really need some understanding of the derivative to get why $e^x$ is a better function to work with than $2^x$.
I guess the example of a bank account continuously compounding interest is a reasonably natural introduction to $e$, but it still revolves around a limiting process that is, at first, counterintuitive and paradoxical.
As a pre-calculus level, I'm worried that the compound interest example serves just as much to demotivate limits (it turns out that the limit isn't infinity - instead, this number you've never heard of that's a little bigger than $2$) as it does to motivate $e$. Also, teenagers don't have bank accounts and tend not to care much about these examples (I have only ever taught them to 18-19 year olds, so I may be wrong, but this has been my experience).
A population doubles every 18 years. Right now it is $1$ million. What is its rate of growth at the present instant?
Another population of $1$ million is currently growing at $20,000$ per year. It continues growing at a rate proportional to its size. How long will it take to double?
These two are just about the simplest examples of problems in which it matters that one particular exponential funtion is "natural".
Well, one thing I've read in a book quite some time ago (unfortunately I don't remember the title, nor the author of that book) is based on information theory. Basically, it is based on the well known Shannon entropy $H=\sum -p_i\ln p_i$. Now the author considered the single terms as the "information contribution" of item $i$. Now $-p_i\ln p_i$ has its maximum at $p_i=1/e\approx 0.37$. Therefore he concluded that if something had a percentage of $\approx37\%$ of a whole, it would stand out. He also noted the closeness to the golden mean (actually to $1/\phi^2$, the length of the shorter line segment when dividing an unit line in the ratio of the golden mean, which is $\approx 0.38$), and speculated that many aesthetic rules usually attributed top the golden mean might instead be connected to this "information maximum".
The Poisson process is a model of occurrences or points that are randomly distributed in time or space. These occurrences could be, for example, emissions of radioactive particles, customers entering a particular shop, fissions of bacteria, or earthquakes in California. All Poisson processes have an exponentially-distributed “waiting time” between events.
If you plot the graph $y=e^x$, choose any point $(x,y)$ and the gradient of the graph at that point will be equal to $x$. Just note that this number is mainly used in statistics (as if you didn't know that already)
