We construct weak solutions of 3×3 conservation laws which blow up in finite time. The system is strictly hyperbolic at every state in the solution, and the data can be chosen to have arbitrarily small total variation. This is thus an example where Glimm's existence theorem fails to apply, and it implies the necessity of uniform hyperbolicity in Glimm's theorem. Because our system is very simple, we can carry out explicit calculations and understand the global geometry of wave curves.