Running EE1 on windows 7 basic. The game crashes immediately after launch once the movies play. I get a message saying that a runtime made the program closed in an unexpected way.

Tried all the compatibility modes. Keep getting the same crash.

If you have integrated Intel video card, which doesn't support hardware transformation and lightening, then i got some bad news for you. It will always crash.
Just like with me. The game is old, but it's require harware t&l - which only some intel gma support.
Here is the notice from the game's page (at the bottom of the infos, just under the system requirements):
Compatibility notice: This game is incompatible with most Intel graphics chipsets. To view a list of Intel chipsets with hardware support for T&L, click here.

If you have another video card (with hardware t&l) then i don't know.

If it is due to the problem that Sydius has described then you could try using a program called 3D Analyze to emulate Texture & Lighting.

Well, I've not used it myself but I have seen it mentioned a few times here on the GOG forums, given as advice for laptop users with integrated graphics who are having video problems. Definitely worth a try. Let us know how you get on.

I tried it with EE, but without any sucsess. Now i will try it with AoE III, which is also requires T&L. If it will run, then the program is good, and the error is "only" EE-related, and i have to use my googling skills only. Which is a good thing, because i googling a lot.
At the moment i waiting for the installer, and collecting my karma points to make sure it will work...
I gonna report in half an hour.

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Well, it's strange. The game starts and runs without the program, but it is slower. Starting it from the program makes it smoth (it's ugly, but i knew that - i only have an integrated card, what else i want?). From this i think the program works, and the problem's root is the EE itself. Now i go and google a little.
(Actually, i could try it out with KotOR 2, but i don't have the discs at hand, so i have to download it from... some bad-bad sites. But i'm not sure i want to do it...).

Report vol 2 SOON (Blizzard-like soon).
(By the way, when i try to run EE, i get this every time: "the instruction at "0x211119d6" referenced memory at "0x0c9c4e88")

Okay, here is the solution (at least for me):
The problem is the screen resolution. Ghilz didn't mentioned it, but i think (and i hope) he has a laptop/netbook with 1024*600 resolution. The game can't handle it, and it will crash with two error messages: first.jpg and second.jpg.
If you have an integrated Intel GMA, then there is the solution (i only tested it with this type of video card). All you have to do is hacking your GMA to make it believe it can run at higher resolutions.

Do this at your own risk! I can't guarantee this will work at your computer (i can only try this at mine, and it's working). Using not supported resolution can damage your hardware.

1)You need a program, called Netbook resolution optimizer, which you can download here.
2) Download it, extract it and start it.
3) Run EE 1 or it's expansion.
Now, here comes the strange thing: you can't see the whole main menu, but you can use your mouse to go to the bottom (like a scroll bar at the side of the screen, but you only have to move your mouse up/down to move the screen. Do not try to change the resolution, it won't work (sometimes causes crash, too). The whol menu is like this , but as soon as you start the game it will work perfectly. At me, it runs at 800x600 and everything is on the screen.
4) Exit the game (only if you want, of course).
5) As you can see, your resolution is 800x600, too.
6) Now right click on the desktop, click "Graphic Properties". It will bring up the Intel control panel (the one you can see on third.jpg). Select "Notebook", then at "Display Setting" you can set the resolution.

As long as the NRC runs, EE will run, too.

I hope it will help because for me, it seolved the problem.

EE, here i come!

Glad you managed to find a solution. If it is indeed resolution based then what would happen if you tried to run the game windowed? Maybe you could then run the window and make it full sized to fit the screen either with a stretch or if the window is sizable.