Although cellular reprogramming enables the generation of new cell types for disease modeling and regenerative therapies, reprogramming remains a rare cellular event. By examining reprogramming of fibroblasts into motor neurons and multiple other somatic lineages, we find that epigenetic barriers to conversion can be overcome by endowing cells with the ability to mitigate an inherent antagonism between transcription and DNA replication. We show that transcription factor overexpression induces unusually high rates of transcription and that sustaining hypertranscription and transgene expression in hyperproliferative cells early in reprogramming is critical for successful lineage conversion. However, hypertranscription impedes DNA replication and cell proliferation, processes that facilitate reprogramming. We identify a chemical and genetic cocktail that dramatically increases the number of cells capable of simultaneous hypertranscription and hyperproliferation by activating topoisomerases. Further, we show that hypertranscribing, hyperproliferating cells reprogram at 100-fold higher, near-deterministic rates. Therefore, relaxing biophysical constraints overcomes molecular barriers to cellular reprogramming. Keywords: Repsox; genomic instability; hypertranscription; p53; reprogramming; single-cell RNA-seq; topoisomerase; transcription factor; transcription rate.