There is none! Variance could be arbitrarily high I suppose. In the limit, if you assume a uniform distribution, I don't think that a single sample will give you any information about the underlying distribution characteristics.

However, for the sake of the argument, variance can be arbitrarily high and still provide increasingly lower amounts of Bayesian evidence. As long as the variance is not unbounded (i.e. as long as the distribution is Gaussian, which I think is a reasonable assumption in nature), you can extract _some_ evidence from a single sample.