How to find the Serial Number for your Chromebook in the software
If your serial number sticker is missing on the exterior of your device, here is an alternative way to discover it in the software.
Step One: Enable Developer Mode
This will wipe your local data, so make sure to back up anything that you don't have stored in the cloud.
To put your Chromebook in Developer Mode:
Press and hold the Esc and Refresh keys together, then press the Power button (while still holding the other two keys). This will reboot your Chromebook into Recovery Mode.
As soon as you see Recovery Mode pop up—the screen with the yellow exclamation point—press Ctrl+D. This will bring up a prompt asking if you want to turn on Developer Mode.
Press Enter to continue, then give it some time. It'll pop up with a new screen for a few moments, then reboot and go through the process of enabling Developer Mode. This may take a little while (about 15 minutes or so), and will wipe your local information.
When it's done, it will return to the screen with the red exclamation point. Leave it alone until it reboots into Chrome OS.
You are now in Dev mode.
On any default screen once in dev mode, access VT2 by hitting 'ctrl+alt+right arrow' [the key above the number 3, or F2 on non-CrOS keyboards]
When it asks for localhost login type root then hit enter
Next, type vpd -l then hit enter
Multiple things will output, but the one you care about is "serial_number" (not "mlb_serial_number" or any other SN reference)
To get out of VT2 and back to the normal OOBE UI, hit ctrl+alt+left arrow (F1)