That which writes down what happened as it happens. A logger's value is that it records without being asked each time.
log (Middle English logge, of obscure origin, tied to the ship's-speed record it produced) with the Old English agent suffix -ere; one who, or that which, keeps the running record.
The watcher that never has to be reminded to write something down — present at every event so nothing depends on someone remembering to note it.