The dynamics of genetic diversity in large clonally evolving cell populations are poorly understood, despite having implications for the treatment of cancer and microbial infections. Here, we combine barcode lineage tracking, sequencing of adaptive clones and mathematical modelling of mutational dynamics to understand adaptive diversity changes during experimental evolution of Saccharomyces cerevisiae under nitrogen and carbon limitation. We find that, despite differences in beneficial mutational mechanisms and fitness effects, early adaptive genetic diversity increases predictably, driven by the expansion of many single-mutant lineages. However, a crash in adaptive diversity follows, caused by highly fit double-mutant 'jackpot' clones that are fed from exponentially growing single mutants, a process closely related to the classic Luria-Delbrück experiment. The diversity crash is likely to be a general feature of asexual evolution with clonal interference; however, both its timing and magnitude are stochastic and depend on the population size, the distribution of beneficial fitness effects and patterns of epistasis.