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So today my League of Legends client crashed at the loading screen for a ranked game. No error messages were displayed. Windows simply told me that Lolclient.exe had stopped responding and closed the program. After several minutes of trying to log back into the game and being told that League of Legends was already running I went in to the process tab of my task manager and found the Lolclient.exe was still running in the background. Once I closed it I was able to log back into the game.
In summery: If your game crashes and you can't log back in, check task manager to make sure it isn't still running in the background!
I hope this helps anyone who has encountered the same problem!