The Lenski experiment investigates the long-term evolution of bacterial populations. Its design allows the direct comparison of the reproductive fitness of an evolved strain with its founder ancestor. It was observed by Wiser et al. (2013) that the relative fitness over time increases sublinearly, a behaviour which is commonly attributed to effects like clonal interference or epistasis. In this paper we present an individual-based probabilistic model that captures essential features of the design of the Lenski experiment. We assume that each beneficial mutation increases the individual reproduction rate by a fixed amount, which corresponds to the absence of epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part of the model. Using an approximation by near-critical Galton-Watson processes, we prove that under some assumptions on the model parameters which exclude clonal interference, the relative fitness process converges, after suitable rescaling, in the large population limit to a power law function.