How can I create a reporting application that collects data from Google Analytics, processes it and returns it for my own custom reporting solution?
It depends on the requirements. You plan to run your own analytics afterwards? Or Archives will be used to predict future trends. As such Google Analytics can do it but if you want to do more with the Analytics data, I guess it is good to archive data on daily basis. You can start here http://code.google.com/apis/analytics/docs/gdata/gdataDeveloperGuide.html
You didn't precise how you would like to collect the data, but I suppose you'll want to run some server script to fetch and process data.
I used the following library to get analytics data from PHP and was quite pleased with it:
http://code.google.com/p/gapi-google-analytics-php-interface/
For Rubyists, I would recommend GARB, which has decent features and traction:
github.com/vigetlabs/garb
Note: add http:// to the link for garb, since I could not post more than one link.