@Nick_Gosse, that's a great idea!
Asana definitely stores completed Tasks. For your specific use case, I'd use Advanced Search to create a search for All Tasks, assigned to you and in the "Completed" state. You could change the completed date ranges as well (to see all Tasks completed in the last week, for example).
I've seen managers use these views to help celebrate all the good work happening across their team, and I use a similar search to help write peer reviews (all completed Tasks where myself and a coworker are followers, to help remind me of the work we collaborated on). Just another reason to track everything in Asana