Tell Windows 7 to Stop Trying to Upgrade to Windows 10?

Further Googling led me to a registry key that looks like it controls the upgrade process.

The machine that was trying to upgrade had these values:

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\WindowsUpdate\OSUpgrade]
"KickoffDownload"=dword:00000001
"KickoffSource"=dword:00000000

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\WindowsUpdate\OSUpgrade\State]
"OSUpgradeState"=dword:0000000e
"OSUpgradeStateTimeStamp"="2015-08-04 15:59:55"
"OSUpgradePhase"=dword:00000002

Another Windows 7 machine, where I had removed KB3035583 before Windows 10 was released (and thus successfully blocked the upgrade), had these values:

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\WindowsUpdate\OSUpgrade]

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\WindowsUpdate\OSUpgrade\State]
"OSUpgradeState"=dword:00000001
"OSUpgradeStateTimeStamp"="2015-07-28 10:09:55"

I updated the registry on the "upgrading" machine to match the registry of the "blocked" machine, then rebooted the machine.

Also, thanks to the comment from @Ramhound, I found a $Windows.~BT hidden folder with 5.1GB of content. I ran Disk Cleanup and removed "Temporary Windows installation files." That folder is now gone.

We'll see if this successfully blocks the upgrade going forward.

Update 7 August 2015

The machine has stopped trying to upgrade to Windows 10, so the registry change must have worked. The values that I set two days ago have not changed.

This is now easy with Never 10, a portable utility to prevent Windows 10 upgrades.