I like user5482's answer, and it is a good one. If you are logged in, and the view is using "rendered entity" instead of "fields", the Edit context link would be accessible from there and you would have your answer in two clicks and no code.
If you truly want one click that goes straight to the edit page, you will need a custom module. You will need to implement hook_menu in your module, and register a static path. In that hook you should define the access callback as using the appropriate drupal permission; I think you want to find the edit instead of the generic access_content.
You will also need to define a custom callback function that will have the dynamic query to find the most recent (published?) node from the database, construct the url, and execute a drupal_goto to take the user there. This way you can have a static address (good for menus) link to a dynamic location (most recent node edit page).
But again, all this custom code work will only save you one click over user5482's answer. Since I cannot think of a reason you would want anonymous users editing nodes, I am guessing this is a menu item for admins only. That extra click shouldn't destroy the ux.