Take $n$ independent uniform random real numbers from $0$ to $e$ and find their product. Do this an infinite number of times and find the median of the products. What does the median approach as $n\to\infty$?
Playing with excel and its random number function, the median of the product does not seem to vary much with $n$. For $n=2$, I worked out (by considering the area under the curve $xy=m$) that the median of the product is the solution to $e^2-3m+m\ln{m}=\dfrac{e^2}{2}$, approximately 1.379. The limit seems to be slightly greater than this, around 1.395.
Here is an article that gives the probability density function of the product of $n$ independent uniform random variables. I've tried integrating the pdf from $0$ to $m$ and setting that equal to $\frac{1}{2}$, without much progress.
Context: no specific context; I just thought of the question and found that I cannot answer it.
My background: high school math teacher. (I suspect that the justification of the answer to my question may be beyond my ability to comprehend, but perhaps others may benefit.)