Darwinian evolution of immunoglobulin genes within germinal centers (GC) underlies the progressive increase in antibody affinity following antigen exposure. Whereas the mechanics of how competition between GC B cells drives increased affinity are well established, the dynamical evolutionary features of this process remain poorly characterized. We devised an experimental evolution model in which we "replay" over one hundred instances of a clonally homogenous GC reaction and follow the selective process by assigning affinities to all cells using deep mutational scanning. Our data reveal how GCs achieve predictable evolutionary outcomes through the cumulative effects of many rounds of imperfect selection, acting on a landscape shaped heavily by somatic hypermutation (SHM) targeting biases. Using time-calibrated models, we show that apparent features of GC evolution such as permissiveness to low-affinity lineages and early plateauing of affinity are best explained by survivorship biases that distort our view of how affinity progresses over time.