Why is $\exp(-1)$ a good benchmark in many physics? In many subjects in Physics (not only phys though), we often encounter a value which gives $\exp(-1)$, such as a time constant in circuitry ($\exp(-t/RC)$, $t=RC$ gives $exp(-1)$), lifetime, mean free path, etc. But I don't quite understand why $exp(-1)$ is a good criterion, because exp(-1) is just ~0.368. A 30% is still an unambiguously large number. If you make a device by believing this value as a cutoff, it might not work as you expected.
So, where is $exp(-1)$ derived from?
 A: Things like particle lifetime should be understood as a rough scale to provide you with intuition about how long you might expect a particle to exist before decaying.  Muons have a mean lifetime of 2.2 $\mu \mathrm s$, but that doesn't mean it would be at all unusual for a muon to decay in $1 \ \mu\mathrm{s}$, or in $4 \ \mu\mathrm s$.  It would, however, be very unusual for one to decay within $1 \ \mathrm{ns}$, or for one to last tens of $\mu\mathrm{s}$.
In essence, many quantities in physics exhibit exponential behavior of the form $e^{-x/A}$, and is a good measure of the scale by which $x$ should be considered large or small.  If $x\ll A$, then $e^{-x/A} \approx 1$, and if $x\gg A$ then $e^{-x/A}\approx 0$.  There's not much more to it than that, and if you want to be more quantitative - for instance, if you want a time $T$ such that you can be 99% sure that any given muon will decay within $T$ - then the mean lifetime of $2.2\ \mu\mathrm{s}$ is not the right number to pick.
A: We use the $1/e$ decay time or rate or whatever because it's the average.
exp(-1) is the mean lifetime
So let's say you start with $N_0$ atoms of some radioïsotope with half-life $T$, then the number of atoms that you have over time is
$$
N(t) = {N_0 \over 2^{t/T}} = N_0 \exp\left(-\frac{t}{T}~\ln 2\right)
$$
and so for example between times $t$ and $t+\mathrm dt$ the number of atoms will (assuming an ideal process) change by
$$
\mathrm dN = - {N_0 \ln 2\over T}\exp\left(-\frac{t}{T}~\ln 2\right)~\mathrm dt,
$$
decreasing proportional to itself.
If we wanted to calculate the mean lifetime we would then maybe write something like $$\tau = \frac{1}{N_0}\int_0^{N_0} \mathrm dN ~t(N) = \frac{1}{N_0}\int_0^\infty \mathrm dt~t~\left(-\frac{\mathrm dN}{\mathrm dt}\right),
$$
the minus sign coming from the change in the direction of integration. If you do the problem straight (per Einstein, "chalk is cheaper than grey matter") it will cancel with the minus sign in the prior equation, or we'll get rid of it here by integrating by parts. We can raise $\mathrm dt~(\mathrm dN/\mathrm dt) \mapsto N(t),$ lower $t \mapsto \mathrm dt,$ call it a day. Thus
$$
\tau = \frac{-1}{N_0} ~ \left.t ~N(t)\vphantom{\int}\right|_0^\infty 
+ \frac{1}{N_0}\int_0^\infty \mathrm dt ~N(t).
$$
The first expression is zero at its lower bound because zero times anything is zero, and zero at its upper bound because the exponential diminishes so much faster than $t$ increases. So ignoring the first term, we get
$$\tau = \int_0^\infty \mathrm dt~\exp\left(-\frac{t}{T}~\ln 2\right) = \frac{T}{\ln 2}.$$
Thus we just get used to writing $e^{-t/\tau}$ where $\tau$ is the mean lifetime. Same with penetration depth, if something penetrates a surface like $\exp(-kx)$ then the mean penetration depth is going to be $1/k.$
So this is something that you’ll just see in a bunch of contexts, using the average to speak of a distribution. And it's measurable!
So some ultraviolet ray penetrates into flesh and you want to know how far it penetrates, it is reasonable to just measure the heating of that flesh as a function of depth and then find the average depth it entered. Sure, some of it went deeper, but the exponential distribution is not "long-tailed" so after a few multiples of this average depth you have the vast majority of the absorption, if you care about that.
exp(-1) is also, unexpectedly, a width of a peak in frequency-space
Similarly for oscillating phenomena, if you have the signal $$
s(t) = \begin{cases}e^{-t/\tau} ~e^{i\omega_0 t},& t\ge0\\0&\text{otherwise}\end{cases}
$$this is a pure sine wave decaying over a time. The exact exp(-1) meaning is not 100% clear and in fact it should probably be defined this way: if you measured its intensity $|s(t)|^2$ you would just get $\exp(-2t/\tau)$ and so the $\tau/2$ is the mean lifetime of the energy in this wave, the exp(-1) target.
This gets really interesting when we transform into Fourier space,
$$
\begin{align}
s[\omega] &\stackrel{\text{def.}}{=} \int_{-\infty}^\infty \mathrm dt~e^{-i~\omega~t}~s(t)\\
&= \frac{1}{1/\tau + i(\omega - \omega_0)}
\end{align}
$$and similarly one would use the absolute square of $|s[\nu]|^2$ to measure energy at different frequencies, giving the classic Lorentzian,
$$ \begin{align}
|s[\omega]|^2 &= \frac{1}{1/\tau^2 + (\omega - \omega_0)^2}\\
&= \pi\tau^2\gamma ~~  \frac1{\pi\gamma\left[1 + \left(\frac{\omega-\omega_0}{\gamma}\right)^2\right]}~,~~\gamma=1/\tau.
\end{align} 
$$[this idiosyncratic way of writing a Lorentzian is because $1/(1 + x^2)$ integrates to $\tan^{-1}(x),$ so that fraction has area-under-the-curve 1.]
So this curve has a full-width-at-half-maximum (FWHM) due to looking where $\frac{\omega-\omega_0}{\gamma} = \pm 1,$ as $\Delta\omega = 2\gamma.$ And so this same energy exp(-1) constant $\tau/2$ is suddenly also a natural measure of the width of a signal in frequency-space. [We cannot use variance like we used mean lifetime before because Lorentzian curves have infinite variance! So $\Delta \omega_{\text{FWHM}}$ is a more neutral measure of width.]
Since the exp(-1) constant is the bandwidth, this then leads to the $Q$-factor, a measure of how many oscillations you get before the oscillator has more or less died out, being described just as $Q = w_0~\tau/2.$ Exercise: for a mass-spring-damping system $m \ddot x + \lambda \dot x + k x = 0,$ derive that $Q\approx\sqrt{mk}/\lambda,$ as $\tau = 2m/\lambda.$ The oscillation is a bit detuned by $\lambda$ so does not happen at $\Omega = \sqrt{k/m}$, but writing $\lambda/m = \Omega/q$ for some $q$ you should be able to find something like $Q = q/\sqrt{1 - 1/(8q^2)}$ or some such, it's been a while so I'm not sure I know the numbers exactly.
Anyway this is another routine measurement type that physicists are making all the time, especially our friends in Optical sub-disciplines, but also like in my side, Condensed Matter. It's just that often you get these nice answers in frequency-space, because you're probing the system with a particular driving frequency that you can sweep over an interval, say... so you get some graph and you want to quickly describe it to someone else, it's really handy to just say "here was the maximum we observed, half of that maximum is so-and-so, here's where the graph hit that so-and-so, and here was the frequency-difference between those two half-maximums." Sure the curve is not necessarily perfectly symmetrical and has parts outside of this half-maximum spread, but it's a handy measurement you can use to describe it, say "we built a resonator with a nanotube over a pit, by driving it with an oscillating gate voltage we observed a $Q$-factor of $10^4$!"
