The fast dynamo problem for steady chaotic flows is approached by evolving magnetic field numerically at zero magnetic diffusion. Two flows are used, an ABC flow and a model flow constructed by patching simple flows together; both flows possess chaotic webs. From smooth initial conditions the magnetic field evolves fine structure in the chaotic regions of the flow because of exponential stretching and folding. Evidence for constructive folding of magnetic field within the chaotic regions is obtained: spatial averages of the magnetic field grow exponentially in time and this process is robust to different methods of averaging. This constructive folding is suggestive of fast dynamo action since the main effect of weak diffusion is to average field locally and smooth out fine variations. The growth and behaviour of initial conditions belonging to different symmetry classes is explored. In the case of the model flow, the dynamo process is explained in terms of chaotic stretching and folding together with cutting and pasting at hyperbolic stagnation points.