How to solve this problem?

Windows 7

The launcher Icon has this administrator icon on it, so if you click on it it will prompt for admin password.

If I uncheck the "Run as administrator" checkbox in the advanced settings of the compatibility tab I can start the game as normal user, but it doesn't start and I get the message, that some files are missing, I should reinstall.

It seems, that the game can't read / write / execute files with a non-privileged user...

How to fix this?

This seems a permissions related issue.
How have you installed the game? Which directory have you installed the game into?

Yes I already wrote that. ;) Like I always install gog-games (over houndred for now): click on the exe that comes with the so called offline-installer, type in admin password, go throught the setup.
Directory is default dir C -> GOG Games

Very well. I suppose that you haven't suffered from the same problem in the past: this is strange if you have always followed the same procedure when installing the game, particularly avoiding to use special directories such as "Program Files" as installation folders.

I'm not familiar with Windows 7, but I think you should take a deeper look at the executable (and/or the shortcut) properties. Right click on the file and open the Security tab: your user should have full control over the executable.

I run a standard user account and ran into this problem as well. The game adds some registry keys to HKEY_CURRENT_USER during installation, and since you must install as admin, it fails to see these keys when ran from your standard account.

To fix it, add these registry keys using your standard account (you can copy these lines to a .reg file and add them):

Windows Registry Editor Version 5.00

[HKEY_CURRENT_USER\SOFTWARE\Paper Sun\Roots]
"Root"="C:\\The Longest Journey\\"
"Sound"="C:\\The Longest Journey\\"
"Fmv"="C:\\The Longest Journey\\"
"Levels"="C:\\The Longest Journey\\"
"Program"="C:\\The Longest Journey\\"
"Global"="C:\\The Longest Journey\\Global\\"
"Static"="C:\\The Longest Journey\\Static\\"

NOTE that you will need to change the path to wherever you installed the game to. I install directly to the C drive so your installation may be different. Remember to escape the backslashes (e.g. C:\\GOG Games\\The Longest Journey\\) in the .reg file if you change the path.