When you use the free upgrade to Windows 10 or pay for a digital retail version, you don’t get a product key. Instead, your computer uses what Microsoft calls a digital entitlement.
Once activated, a digital entitlement for your Computer's hardware is created in the Microsoft Windows 10 activation service.This entitlement can be used by the same Computer again for re-activation of the same edition of Windows 10 in the future.
What all this means is when you are reinstalling Windows 10 on the SAME computer, Windows does not require a PRODUCT KEY but will instead connect to the Microsoft Windows 10 Activation Server and automatically activate.
If you are installing Windows 10 for the first time you will of course require the Product Key to activate Windows 10. But subsequent reinstalls should not require a product key as Microsoft remembers your computer. When installing Windows 10 and it asks you for a product key, just click on Do This Later or Skip.
This is also true if you upgraded from Windows 7 or Windows 8 to Windows 10.
If your system originally came with Windows 7 or 8 and you upgraded the system to Windows 10 then you won't have the original Windows 10 install disk to do a reinstall, but instead you will need a Windows install image, available from Microsoft HERE and once installed your system should automatically activate.
There will of course be exceptions to this, especially if you change your hardware. Installing a new hard drive or RAM will most likely not affect this, but changing the motherboard will.