I've always thought that Opera is the browser most closely related to Google's philosophy. It may seem strange, but Opera's mail client has many things in common with Gmail (labels, a threaded view), Opera includes search in most views and it usually adds innovative features long before the competition. Another interesting tidbit is that Opera has been supported by Google's contextual ads.
Unfortunately, many Google services don't work in Opera, because of some browser flaws or lack of testing. Another reason is that Opera has a very small market share and very few people care about supporting it.
Opera 9.5, launched today in alpha, tries to solve two of the most important Google problems: persistence of your browser's settings and a searchable web history. Google already has solutions for these problems, but they're limited and not properly integrated. Google Browser Sync is a Firefox extension that synchronizes your bookmarks, cookies, passwords and sessions across multiple computers, while Google Toolbar lets you send all the pages you visit to Google Web History, where you can search them.
Opera also makes your history searchable, but it stores the index on your computer, the same as Google Desktop. You can search the full content of the pages, from the History page or from the address bar.
"With Opera 9.5, we introduce full history search - search the complete Web pages you visited, but without requiring you to bookmark them. Unlike previous and other browser history searches which only look at the URLs of the pages you have visited, full history search searches the actual content of the Web pages you have visited."
If you have a My Opera account (maybe you're a fan), you can store your bookmarks online and have them automatically synchronized every time you make a change. There's no interface for viewing the bookmarks online and no other data is synchronized for the moment. The feature will become even more valuable when you'll be able to synchronize your bookmarks between the desktop version and Opera Mini or Opera for Wii.
So I think it's ironic that the software most closely related to my definition of a Google Browser doesn't work well with many Google services, including Google Docs, Picasa Web Albums, Google Notebook, Google Page Creator, Blogger (this script solves some of the problems).
The latest build of Opera 9.5 is available here for Windows, Mac and Linux. Don't install it over a previous Opera installation or if you're not adventurous.