I've got pretty much complete freedom to work on what I want as long as it's related to Search and will benefit the company if it succeeds. I came into Google with some minor name recognition in the startup & Haskell communities (I wrote a top Haskell tutorial and a blog about my experiences founding a failed startup) but no world-changing accomplishments, and between then and getting my current project I contributed to 4 major "wins" in search (Search Options, moving the individual property searches to www.google.com, the visual redesign of 2010, and the Authorship program), plus some pinch-hitting on the Google+ and Doodle teams.
I think this type of freedom does differ among departments. G+, Docs, and Android are very command-and-control, with somebody at the top deciding exactly which features everyone will be working on and relatively little individual discretion for engineers. Research is the complete opposite end of the spectrum - usually researchers are brought into Research, and then they keep doing their specialty but with more data available. Search, Chrome, YouTube, GMail, and Maps all give engineers a large amount of discretion in proposing projects from the ground up and then working on them.