Where to find windows product key "other than settings within windows"?

1. You didn't tell us what version of Windows. ( I realize you posted in the Win10 forum, but people here often post in the wrong forum so I don't assume.) If Windows 10 was installed on that computer and was legally activated, you can reinstall Windows 10 anytime and it will activate automatically. If it was Windows 8.1 I think the same may have been true. I'm not sure about Win8, and Win7 definitely didn't activate automatically.
2. If you still have access to the Windows system on a drive, even if the drive is not installed, you can find the product key by browsing the system even from Linux. If you took a system image of the drive with your Windows system, you could restore that image to another drive to browse it.
3. If it was Windows 7 then the product key was probably on a sticker on the computer case or, if a laptop, possibly in the battery bay. I think Windows 8 may have done the same and I think perhaps Windows 8.1 was when Microsoft stopped doing that. Windows 10 definitely had no physical product key stickers.

Generally, if you bought a physical copy of Windows, the product key should be on a label or card inside the box that Windows came in. If Windows came preinstalled on your PC, the product key should appear on a sticker on your device. If you've lost or can't find the product key, contact the manufacturer.