How to install Chrome for just the current user?

You might consider using the Portable Apps version of Chrome.

http://portableapps.com/apps/internet/google_chrome_portable

This version does not require "installation" and its updates are maintained by the PortableApps team instead of Google. I use it to have a 2nd version of chrome for testing.

You can use Google Chrome portable, unzip it to a folder you have access to, and create a shortcut to it to where you want it to be (desktop, start menu, pin to taskbar, etc)

Because it is portable, it is not installed. Because you make the shortcut where you want it to be, no one else sees it. If someone still deletes Google Chrome, they're actively doing so because they don't want the software to be on your pc. If you are at a work location, it is likely this is against a policy, and you should contact a system administrator and ask if Google Chrome is allowed.