At my last job I was a web programmer, PPC manager, SEO expert and Analytics expert (jack of all trades - master of none?).
One of the things I'd do would be to find pages with above average exit rates and try to figure out why. I'd take a look at how people were getting to the page (search terms, external link text, internal link text) and then look at the page itself and see if it had the information users might expect.
I'd also do evaluations of online marketing campaigns that companies were using. We had one large university as a client and they had been spending $250/month for 2 years for a banner at a site that wasn't even sending traffic. I then showed their marketing department how they could use the client software we gave them to track campaigns.
Another thing I would set up would be reports for goal pages. For example lets say a form to request more information was a goal. I'd give them a report showing them labels for all visitors that filled in the form. Then they'd know if google was working better than yahoo, if visitors from a campaign on site X were signing up, which PPC campaigns were performing better than others, etc.
We used ClickTracks Pro for our analytics engine. So depending on the client we either gave them the client browser so they could do their own reports or we would do reports for them. Obviously there was a lot more analysis than what I listed above and once the data was evaluated it could require changes from different departments to get a solution implemented. It might require a PPC ad change, it might require a link text change on an advertisers site, it might require a design change to place a link in a more prominent position, content changes on a page to meet users expectations or improve SEO results, etc..