How can I tell when the last time Chrome updated?
If you're using Windows 7 or Vista (maybe XP, but don't have it handy to test) Chrome actually installs to AppData\Local\Google\Chrome\Application which will contain the previous version and the current version along with sub-directories named after the version number.
The directory names and date information hold the information you asked for.
Update: Chrome often installs under Program Files now so the new location to check would be: C:\Program Files (x86)\Google\Chrome\Application
In OS X, open a Terminal and enter:
ls -l "/Applications/Google Chrome.app/Contents/Frameworks/Google Chrome Framework.framework/Versions"
This will print the current version and the previous one (including its installation date):
drwxrwxr-x 11 werner admin 352 Aug 24 01:13 76.0.3809.132
drwxrwxr-x 11 werner admin 352 Sep 18 01:03 77.0.3865.90
lrwxrwxr-x 1 werner admin 12 Sep 18 01:03 Current -> 77.0.3865.90