We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate, and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is included. We find a strikingly high rate of entropy production that would limit information representation in cortical spike patterns to the immediate stimulus response.