Just bought and installed a copy of the first Empire Earth game. When I start it, the opening videos play fine but it then ends up on a black screen with music playing and I have to Ctrl+Alt+Del to get out of it.

I have tried changing the compatability settings, making sure to run it as an administrator and all the other things I usually do when a game from gog doens't work, but no success this time.

I'm running a laptop with Windows 7 64 bit and have an nvidia card.

Please help, as this purchase is starting to look like one I'll regret...

Maybe some older components of DirectX9 are missing on your Win7 PC.
Install the DirectX End-User Runtimes to complete your system:
http://www.microsoft.com/en-us/download/details.aspx?id=8109
Run the file and extract the content to a separate folder. Then run dxsetup.exe to start the installation.

Thanks, I might try this later, but after a few more attempts it looks like this might be a uac problem. Sometimes the game will start but after loading/starting a new game or trying to quit, the screen will go black and when trying to bring up the task manager there will a prompt asking for permission to run the exe again. Saying yes does nothing.

Does anybody know a solution to this that doesn't involve me turning off uac?

Try running it explicitly as administrator, by right-clicking the game's shortcut and selecting "Run as administrator".

It could also be a Data Execution Prevention (DEP) issue. Add the game to DEP's exception list (see here).