I don't know of a plugin that exists expressly for this purpose, but you could combine LumberJack (logging plugin) with Craft's numerous events to make this happen with your own plugin. You can log from within said plugin or even in your Craft templates.
Take a look at the event reference and see if there's enough there to satisfy your client's needs. Creating theyour plugin should be pretty straightforward.
It may also be worth noting that Craft's logs are available from the control panel at /admin/utils/logs
. These are developer-centric errors and warnings though, not a record of specific user actions.