DNA performs diverse functional roles in biology, nanotechnology and biotechnology, but current methods for autonomously synthesizing arbitrary single-stranded DNA are limited. Here, we introduce the concept of primer exchange reaction (PER) cascades, which grow nascent single-stranded DNA with user-specified sequences following prescribed reaction pathways. PER synthesis happens in a programmable, autonomous, in situ and environmentally responsive fashion, providing a platform for engineering molecular circuits and devices with a wide range of sensing, monitoring, recording, signal-processing and actuation capabilities. We experimentally demonstrate a nanodevice that transduces the detection of a trigger RNA into the production of a DNAzyme that degrades an independent RNA substrate, a signal amplifier that conditionally synthesizes long fluorescent strands only in the presence of a particular RNA signal, molecular computing circuits that evaluate logic (AND, OR, NOT) combinations of RNA inputs, and a temporal molecular event recorder that records in the PER transcript the order in which distinct RNA inputs are sequentially detected.