Chaotic signals have emerging applications which include masking, secure communication and dynamical modeling of biological systems. However, no general methodology for synthesizing chaotic oscillators presently exists. Here, it is shown that chaotic oscillators can be obtained through genetic optimization of the topology of bipolar transistor-based circuits aimed at maximization of approximate signal entropy. A selection of circuits were physically realized and characterized, with features resembling previously-described chaotic oscillators that had been designed by empirical modification of traditional oscillators or implementation of differential equation systems known a-priori to be chaotic. The proposed approach offers a novel means to generate chaotic signal sources without prior topological assumptions, and the emergence of chaos from signal entropy maximization has potential implications for the study of biological neural networks.