In rapidly adapting asexual populations, including many microbial pathogens and viruses, numerous mutant lineages often compete for dominance within the population1-5. These complex evolutionary dynamics determine the outcomes of adaptation, but have been difficult to observe directly. Previous studies have used whole-genome sequencing to follow molecular adaptation6-10; however, these methods have limited resolution in microbial populations. Here we introduce a renewable barcoding system to observe evolutionary dynamics at high resolution in laboratory budding yeast. We find nested patterns of interference and hitchhiking even at low frequencies. These events are driven by the continuous appearance of new mutations that modify the fates of existing lineages before they reach substantial frequencies. We observe how the distribution of fitness within the population changes over time, and find a travelling wave of adaptation that has been predicted by theory11-17. We show that clonal competition creates a dynamical 'rich-get-richer' effect: fitness advantages that are acquired early in evolution drive clonal expansions, which increase the chances of acquiring future mutations. However, less-fit lineages also routinely leapfrog over strains of higher fitness. Our results demonstrate that this combination of factors, which is not accounted for in existing models of evolutionary dynamics, is critical in determining the rate, predictability and molecular basis of adaptation.