Google's Chrome browser does collect user data and send it back to Google.

Google has admitted the auto-suggest feature of Chrome's Omnibox gives it potential access to users' keystrokes, providing the company with a wealth of information on the browsing habits of its users.
"When you type URLs or queries in the address bar, the letters you type are sent to Google so the Suggest feature can automatically recommend terms or URLs you may be looking for"
Unlike searching through Google, so long as the user has auto-suggest enabled, and Google set as their default search engine, the Omnibox will grant Google access to search enquiries without the user ever hitting the enter key.

Google says it intends to store about 2% of this information, alongside the IP address of the computer that typed it.

So, if you don't want that to happen, just disable the autosuggest feature and don't have Google as your default search engine