Is there a way to log what layouts gets used/accessed during a time period?

I guess that if the user navigates with custom scripted buttons only, you could create a script that writes to a log table.

And each navigation scripts should call this script at the end.

It is what I did for a project. At each script I called the "writeToLog" script.

There was a session at DevCon, "Build Better Solutions by Tracking User Activity," that demonstrated using a tiny web viewer on each layout, linked to a web page which was linked to Google Analytics.