It is well-known that parameter estimates and forecasts are sensitive to assumptionsabout the tail behavior of the error distribution. In this paper we developan approach to sequential inference that also simultaneously estimates the tailof the accompanying error distribution. Our simulation-based approach modelserrors with a t -distribution and, as new data arrives, we sequentially computethe marginal posterior distribution of the tail thickness. Our method naturallyincorporates fat-tailed error distributions and can be extended to other datafeatures such as stochastic volatility. We show that the sequential Bayes factorprovides an optimal test of fat-tails versus normality. We provide an empiricaland theoretical analysis of the rate of learning of tail thickness under adefault Jereys prior. We illustrate our sequential methodology on the Britishpound/US dollar daily exchange rate data and on data from the 2008-2009credit crisis using daily S&P500 returns. Our method naturally extends tomultivariate and dynamic panel data.