How to disable the "Get Windows 10" icon shown in the notification area (tray)?

If you just want to remove the tray icon until the next restart you can terminate the GWX.exe process using Task Manager.

To get rid of the icon permanently, uninstall KB3035583 which is responsible for these notifications: Control panel, windows update, installed updates, sort by name, "Update for Microsoft Windows KB3035583" (not a Security Update), uninstall, reboot.
(Alternative: open CMD and enter wusa /uninstall /KB:3035583)

When you're offered the same again via Windows Update remember to hide it.

After uninstalling, if remnants of the update's files are still in Windows\System32\GWX, just delete that directory, although first you may need to take ownership of it.

• Run Autoruns as Administrator, unhide Windows/Microsoft entries via Options
• Search for gwx.
• Disable the items that don't give an access denied message.

According to TechJourney, you can make a small registry change to prevent the application starting.

This registry key will stop Gwx from starting at boot:

[HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\Gwx]
"DisableGwx"=dword:00000001

To create this:

1. Run regedit.exe as an administrator.
2. Create a key named Gwx within HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\.
3. Create new dword value, named DisableGwx, with a value of 1.

Uninstalling and blocking the update from the system (as in other answers) is certainly a cleaner approach, if you don't intend to use it.