Studying cellular and developmental processes in complex multicellular organisms can require the non-destructive observation of thousands to billions of cells deep within an animal. DNA recorders address the staggering difficulty of this task by converting transient cellular experiences into mutations at defined genomic sites that can be sequenced later in high throughput. However, existing recorders act primarily by erasing DNA. This is problematic because, in the limit of progressive erasure, no record remains. We present a DNA recorder called CHYRON (Cell History Recording by Ordered Insertion) that acts primarily by writing new DNA through the repeated insertion of random nucleotides at a single locus in temporal order. To achieve in vivo DNA writing, CHYRON combines Cas9, a homing guide RNA and the template-independent DNA polymerase terminal deoxynucleotidyl transferase. We successfully applied CHYRON as an evolving lineage tracer and as a recorder of user-selected cellular stimuli.