UPDATE 12/7/09: We believe this issue has been resolved. If you changed your configuration files as a workaround to this problem, please restore the old files or reinstall the game.
Many players in Europe are experiencing problems opening League of Legends. We're aware of the issue and working to resolve the problem.
If you're experiencing this issue, please attach your patcher_lib.log to a post in this thread.
Warning about workaround fix
Some players find that editing their internal League of Legends files resolves the issue. I can't recommend this, even though it appears to be a consistent workaround. What you're effectively doing is tell the launcher not to check for updates in the future. This works right now because you have an updated version, but when a patch is released, you may not be able to update. You will either need to restore the launcher config to the way it was, or repair/reinstall the game.
If you choose to do the workaround posted on page 2, please make a backup of the launcher_config.xml and restore the file after this issue has been resolved.