session-recorder (formerly swank-recorder) is a simple library to record input sent to Lisp via Slime or Sly. It records the internal-real-time and the string of code to a file of your choosing. Just load the system, call (start-recording "/path/to/your/output/log.lisp") and go! A stop-recording function is provided, but not strictly needed since each output is flushed immediately.
Playback of recordings is also in the works, though not fully functional at the moment yet.