Gmail's help center lists the supported browsers and includes the minimum version number required next to each browser, except Google Chrome.
Unfortunately, even if Chrome installs the latest version, you can't use it until you restart the browser. Google started to add a little orange dot next to the wrench icon and show messages like: "Old school's not cool. Google Chrome is woefully out of date because it hasn't crashed or restarted in a while. Restart Chrome to apply update."
{ The second screenshot is licensed as Creative Commons by Chris Messina. }