The exponential growth in the rate at which information can be communicated through an optical fibre is a key element in the 'information revolution'. However, as for all exponential growth laws, physical limits must be considered. The nonlinear nature of the propagation of light in optical fibre has made these limits difficult to elucidate. Here we use a key simplification to investigate the theoretical limits to the information capacity of an optical fibre arising from these nonlinearities. The success of our approach lies in relating the nonlinear channel to a linear channel with multiplicative noise, for which we are able to obtain analytical results. In fundamental distinction to linear channels with additive noise, the capacity of a nonlinear channel does not grow indefinitely with increasing signal power, but has a maximal value. The ideas presented here may have broader implications for other nonlinear information channels, such as those involved in sensory transduction in neurobiology. These have been often examined using additive noise linear channel models but, as we show here, nonlinearities can change the picture qualitatively.