Using climate change as a prototype motivating example, this paper analyzes the implications of structural uncertainty for the economics of low-probability high-impact catastrophes. The paper shows that having an uncertain multiplicative parameter, which scales or amplifies exogenous shocks and is updated by Bayesian learning, induces a critical "tail fattening" of posterior-predictive distributions. These fattened tails can have strong implications for situations (like climate change) where a catastrophe is theoretically possible because prior knowledge cannot place sufficiently narrow bounds on overall damages. The essence of the problem is the difficulty of learning extreme-impact tail behavior from finite data alone. At least potentially, the influence on cost-benefit analysis of fat-tailed uncertainty about the scale of damages -- coupled with a high value of statistical life -- can outweigh the influence of discounting or anything else.