Cauchy distribution. https://en.m.wikipedia.org/wiki/Cauchy_distribution The mean of a sample of cauchy(m,s) variables has that same cauchy(m,s) distribution, regardless of sample size.

Meaning to matter how many samples you take, the mean (of the samples) is just as variable as an individual sample.

The mean and variance of the distribution (equivalently, of infinite samples) are undefined. The Cauchy is equivalent to a t distribution with 1 degree of freedom.

Infinite variability will be undesirable for lots of reasons though.