Viral showed me a striking result about the distribution of the error introduced in adding up M floating point numbers which are drawn from the normal distribution.
That got us talking about the Cauchy distribution (also known as the t distribution with 1 degree of freedom). This is an infinite variance distribution, so the sample mean is not consistent. It is a striking example to challenge our everyday intuition which thinks the mean is always consistent. So I added an example about the Cauchy distribution to my page of R examples. Try it - it's fun! :-)