Sends the name of the file you're downloading to Google for whitelist checking; stores your IP address associated with the file for a few weeks.[1]
Every URL you even begin to type in the address bar is sent to Google, in whole or in fragments, for auto-completion purposes.[2][3]
Connects to Google every 30 minutes to download a list of malicious URLs, so the fact that you even have Chrome open is transmitted to Google.[1]
Asks you to login to your Google account, so your browsing tabs, history, etc. is stored on Google servers.[4]
Connects to websites in the background before you are even finished typing them in, without your explicit instruction.[5]
Contains an RLZ identifier, an encoded string sent together with all queries to Google.[6]
Summary: There is nothing, nothing, you can do in Chrome that isn't transmitted to Google through some channel.
Welcome to the botnet.
Recommended Alternatives
GNU IceCat A fully free, libre, open-source (FLOSS) fork of Firefox without its built-in DRM support and Firefox Hello. Comes with LibreJS, an add-on which blocks out any non-free javascript on webpages.
Pale Moon A fork of Firefox optimized for modern CPU architectures. Does not include Firefox's Australis GUI.
Qupzilla A WebKit based browser aimed towards GNU/Linux users. Sports a cleaned up GUI that integrates better with most desktop environments, a built in ad blocker, and a unified history/bookmarks/RSS reader window.
Midori A lightweight WebKit based browser. Features a built-in ad blocker and cookie manager.
If you're thinking of switching to Chromium (Google's open source version of Chrome) instead, think again. It still phones home to Google,[9] and for some time, snuck in a binary blob for voice recognition whenever you started it up.[7] This has been removed by the project maintainers,[8] but it should raise some concern over what else could be lurking in the code.
However, if you absolutely must use a Chrom* based browser, Iridium would be the best choice, as it is a FLOSS fork of Chromium stripped of as many botnet features as possible.