What processes could generate Laplace-distributed (double exponential) data or parameters? Lots of distributions have "origin myths", or examples of physical processes that they describe well:


*

*You can get normally distributed data from sums of uncorrelated errors via the Central Limit Theorem

*You can get binomially distributed data from independent coin flips, or Poisson-distributed variables from a limit of that process

*You can get exponentially distributed data from waiting times under a constant decay rate.


And so on.
But what about the Laplace distribution?  It's useful for L1 regularization and LAD regression, but it's hard for me to think of a situation where one should actually expect to see it in nature.  Diffusion would be Gaussian, and all the examples I can think of with exponential distributions (e.g. waiting times) involve non-negative values.     
 A: Define a compound geometric distribution as the sum of $N_p$ iid random variables $X_N = \sum_i^{N_p} X_i$, where $N_p$ is distributed like a geometric distribution with parameter $p$. Assume that the iid random variables $X_i$ have finite mean $\mu$ and variance $v$.
It was shown by Gnedenko that in the limit $p\to 0$, the compound geometric distribution approaches a Laplace distribution. 
$Y:= \lim_{p\to 0} \sqrt{p} (X_N - N_p\mu) = Laplace(0,\sqrt{\frac{v}{2}})$
The density of the $Laplace(a,b)$ is $\phi(x) = \frac{1}{2b} \exp\left( - \frac{|x-a|}{2b}\right)$
B.V Gnedenko, Limit theorems for Sums of random number of positive independent random variables, Proc. 6th Berkeley Syposium Math. Stat. Probabil. 2, 537-549, 1970.
A: At the bottom of the Wikipedia page you linked are a few examples:


*

*If $X_1$ and $X_2$ are IID exponential distributions, $X_1 - X_2$ has a Laplace distribution.

*If $X_1, X_2, X_3, X_4$ are IID standard normal distributions, $X_1X_4 - X_2X_3$ has a standard Laplace distribution. So, the determinant of a random $2\times 2$ matrix with IID standard normal entries $\begin{pmatrix}X_1 & X_2 \\\ X_3 & X_4 \end{pmatrix} $ has a Laplace distribution.

*If $X_1, X_2$ are IID uniform on $[0,1]$, then $\log \frac{X_1}{X_2}$ has a standard Laplace distribution. 
