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