How to fix an iPhone stuck on "Connect to iTunes" screen after iOS8 update?
Your device seems to be in Recovery Mode (or DFU Mode but that's hardly the case unless you were jailbreaking). If the firmware update was successful, you might get out of Recovery Mode by simply rebooting the iPhone. iPhone in Recovery mode should also reboot by itself after 15 minutes.

A manual reboot is done the usual way, by holding down the Sleep button and swiping the 'Slide to power off' slider when it appears. After iPhone has shut down, start it by pressing the Sleep button.

If the iPhone stays it the Recovery Mode after the reboot, it must be restored with iTunes. You only option is to find a computer with iTunes and restore the firmware when connected to iTunes. You will lose all non-backed up changes on the iPhone when restoring! After restoring a functional iOS, you can restore the latest backup of your iPhone.

There are also various pieces of software (i.e. TinyUmbrella) that have the means of getting an iPhone out of the Recovery Mode, if the iOS is otherwise functional. However, I'm not aware of such programs available on a Chromebook.

When you get to a computer with iTunes, there's an Apple's support article listing the steps to resolve most iOS update and restore errors with iTunes.


  
  Update iTunes
  Update the OS X
  Update, disable or remove your computer’s security software
  Unplug extra USB devices
  Restart
  Try again