Although discrete mixture modelling has formed the backbone of the literature on Bayesian density estimation, there are some well-known disadvantages. As an alternative to discrete mixtures, we propose a class of priors based on random nonlinear functions of a uniform latent variable with an additive residual. The induced prior for the density is shown to have desirable properties, including ease of centring on an initial guess, large support, posterior consistency and straightforward computation via Gibbs sampling. Some advantages over discrete mixtures, such as Dirichlet process mixtures of Gaussian kernels, are discussed and illustrated via simulations and an application. © 2014 Biometrika Trust.