Genetically encoded DNA recorders noninvasively convert transient biological events into durable mutations in a cell's genome, allowing for the later reconstruction of cellular experiences using high-throughput DNA sequencing. Existing DNA recorders have achieved high-information recording, durable recording, prolonged recording over multiple timescales, multiplexed recording of several user-selected signals, and temporally resolved signal recording, but not all at the same time. We present a DNA recorder called peCHYRON (prime editing Cell HistorY Recording by Ordered iNsertion) that does. In peCHYRON, prime editor guide RNAs (pegRNAs) insert a variable triplet DNA sequence alongside a constant propagation sequence that deactivates the previous and activates the next step of insertion. This process results in the sequential accumulation of regularly spaced insertion mutations at a synthetic locus. Accumulated insertions are permanent throughout editing because peCHYRON uses a prime editor that avoids cutting both DNA strands, which risks deletions. Editing continues indefinitely because each insertion adds the complete sequence needed to initiate the next step. Constitutively expressed pegRNAs generate insertion patterns that support straightforward reconstruction of cell lineage relationships. Pulsed expression of different pegRNAs enables the reconstruction of pulse sequences, which may be coupled to biological stimuli for temporally-resolved multiplexed event recording.