Do you mean your Mac user account does not have admin privileges?
In that case have you tried just installing Chrome into ~/Applications instead?
In that case have you tried just installing Chrome into ~/Applications instead?
So Google Chrome.app resides in /Applications, but it's installed by the account that gets created when you reinstall OS X (which is by default an account with full privileges). Standard users can read /Applications, so they can launch Google Chrome.app, but the process can't write to /Applications because of its effective UUID. That's why it prompts for an admin password.