  We investigate a recursively generated sequence of random variables that
begins with an Exponential random variable with parameter (i.e., inverse-mean)
1, and continues with additional Exponentials, each of whose random parameter
possesses the distribution of the prior term in the sequence. Although such
sequences enjoy some (limited) applicability as models of parameter
uncertainty, our present interest is primarily theoretical. Specifically, we
observe that the implied sequence of distribution functions manifests a
surprising connection to two well-known mathematical quantities first studied
by Leonhard Euler: the Euler-Gompertz and Euler-Mascheroni constants. Through a
close analysis of one member of this distribution-function sequence, we are
able to shed new light on these constants.
