I added my votes to http://checkvist.uservoice.com/forums/2121-general/suggestions/14940-recent-activity-page which appears to be identical.

Definitely these suggestions should be joined.
And, of course you may provide activity report in pro version only. That might be good point to use this app "professionally" - time sheets, TPS reports, the ususal corporate stuff.

Use Case:
I need to see what I have accomplished to transcribe into a report in another application
1) Go to Checkvist
2) click something so that I can see recently finished tasks, probably everything since the last time I did this
3) retype the list into another report.
3a) copy and paste might be helpful, but is not essential because I always end up rewriting the language of my items when I report on them

Suggested UI:
1. in "your lists" at the top, there is a "completed tasks" item
2. Clicking it shows all completed tasks across all lists, in their hierarchy, with the original list at the root item
3. There is a filter control at the top to set start and end dates. it defaults to [-1 week] to [now]