What is the physics problem whose answer is $e$ in the power of $\pi$ despite all given numbers being unity? A few years ago my school physics teacher told us, his students, that there is an elegant physics problem in which all given numbers are unity, but the answer is $e$ in the power of $π$, that is, 

$e^\pi$,

where e = 2.718... is the base of the natural logarithm and π = 3.14... is pi, the constant commonly defined as the ratio of a circle's circumference to its diameter.
Fast forward to now, I am a university student studying something unrelated to physics, and I recently mentioned the above recollection of mine in a conversation with a physicist whom I had helped improve his English in his physics articles. Long story short, he highly doubts that such a physics problem exists. He believes that either I misunderstood my school teacher or the formulation of the problem is too lengthy or unnatural. But I clearly remember my teacher's words that the formulation of the problem is simple and that the solution is elegant. Unfortunately, I do not know what that problem is about. I tried googling, but did not get any lead.  
What is that mysterious problem, or is there any physics problem matching the description above?
 A: The question posted in the existing answer is almost certainly the one your teacher was referring to. For the sake of illustration I'll give another question that works.

There are $N$ identical tiny discs lying on a table with total mass $M$. Another disc of mass $m$ is very precisely aimed to bounce off each of the discs exactly once, then exit opposite the direction it came. Neglect any collisions between the discs themselves.

In the limit $N \to \infty$, what is the minimal value of $M/m$ for this to be possible? Given this value, what is the ratio of the initial and final speeds of the disc?

The answer to this question is $e^{\pi}$. Notably, you don't even need to set any adjustable parameters to one by hand, like you have to set $\mu = 1$ in the capstan problem. In this problem, $M$ is arbitrary, $N$ is taken to infinity, and $m$ is fixed given $M$.
The link between these problems is that the $e$ comes from some sort of exponential decay, while the $\pi$ comes from the fact that there are $\pi$ radians in a semicircle. (The problem is solved by considering the angle through which the velocity of the mass $m$ must turn, which is $\pi$.)
A: $e^\pi$ is the square root of the ratio of the load tension to the hold tension for a rope wrapped $1$ full turn around a capstan when the coefficient of friction is $1$.
If I am allowed to say “$1$ half-turn” without violating the “unity” requirement then taking the square root is unnecessary.
The relationship is
$$T_\text{load}=T_\text{hold}\,e^{\mu\phi}$$
where $\mu$ is the coefficient of friction and $\phi$ is the wrapping angle, as explained in the Wikipedia article “Capstan equation”.
A: Incomplete playing detective. Not a physical example, but a mathematical one from which it might be possible to generalize to several physical problems. 
If you have a rate of change of some quantity proportional to that quantity, i.e. $\frac{dx}{dt}=kx$, then $x=c_0e^{kt}$, where $c_0$ is the initial value of x. 
If $c_0=1$ then one second later, $x=e^\pi$.
So what could make $k=\pi$? $k$ can be the arc length of a semi-circle, the area of a unit circle, or the volume of a cylinder having unit radius and height. 
So if a rate of change is proportional to the present value of the quantity and one of these physical characteristics we should have $e^\pi$. 
Area is a bottle neck for several processes. Chemical reactions, bacteria growth, heat transfer. 
