1st. Just run the script editor and paste the following content:
do shell script "open -a /Applications/Google\\ Chrome.app --args --incognito"
tell application "Google Chrome"
close windows
make new window with properties {mode:"incognito"}
activate
end tell
2nd. Save the above content as a file with the “.app” extension such as test.app.
3rd. Run the test.app you will get a google chrome instance runing with private mode or incognito mode automatically.
Ok, just enjoy it!