I have had this exact same problem. I have verified that my internet is working correctly, by playing games on other accounts and other servers (which worked fine). Since it works on the same computer with different accounts, and EUW clearly isn't down as I see other people playing and the server status board shows that it's up, it seems to be an account- or gamebound issue. The bug is hard to reproduce for me since it only happens very occasionally - the last time was a month ago - but so far it has only happened on EUW. When this happens, I get the "The game has crashed" message that is pictured in the OP, and it stays this way for a couple of hours (probably until the game "times out" on the server). This has been happening sporadically but rarely over a period of several months now. Afterwards, the game disappears from match history as if it never happened. It also appears that no one is able to connect to the game, since when this happens while I play with friends, they get the same problem. Once it happened with someone from a different part of the world, so it's not just a country-wide internet brown-out either. Since it happened just now, I can try to contact my would-be teammates, and see if they got the bug too. I can also provide the summoner names of these people if Riot wishes, in order to help with the troubleshooting, but I do not wish to post them publicly.
If you look at the ingame info on op.gg, it appears that the game "lasts" as long as it takes for the client to reset to normal and let you play again. This is why I say it seems like the game has to time out on the server before things start working again. I've seen posts mentioning "disappearing games", but I'm not sure if this is that same problem.
Because it is a rare issue for me, it doesn't really irritate me that much, but some days when I want to play for a few hours it can be quite annoying when I get "locked out" for all those hours. I have not gotten this while playing ranked yet, so I can't say if it will end a series or make you lose LP. What I can say is that it does not appear in the Match History afterwards, so it seems likely that the game de facto never happened.
So, in the interests of getting this fixed, I will describe the bug with as much detail as I can:
Entering the matchmaking queue, accepting the match, selecting a champion, summoner spells and locking is problem-free. Then, as the game is about to begin, the screen goes black for longer than usual (for me, it's usually only black for a second or so. In this case, it's much closer to a minute) and the "Working..." cursor is displayed. After about a minute (the screen is still black), I get the Error sound from Windows, but nothing is displayed yet. I have to tab out of the game and then click on the game on the start menu, as it is not possible to alt-tab back into the game at this point (it is not displayed in the list of programs available for alt-tabbing), to see the error message, which is exactly as pictured in the OP. Pressing "Retry" results in a repetition of the aforementioned events, from the black screen onwards. Pressing cancel returns you to the launcher, which displays the Reconnect button. The first time I saw this I was intrigued by the HTML code in the error message, as it makes it seem like the error is not meant to be displayed in a message box of that type, but rather either on a webpage or in an Adobe AIR-style message box. This may be a clue to help you figure out what the issue actually is? At any rate, after this point, giving up and waiting for the timeout is the only solution. Resetting the internet connection, relogging, rebooting (both the computer and all network hardware), recycling, switching computers, switching networks, flushing DNS, acquiring a new DHCP lease and many other things have been tried to no avail. Hence it is likely a game-specific server-side problem.
To the OP: It might be a good idea to edit this post to be in accordance with the guidelines mentioned in This post to make it more likely to be visited by a red :3!
Edit: OH! And interestingly, exiting the client gives the "If you exit the game now, you'll receive a time penalty..." (or however it is worded) message, just as if you were closing the client while in champion select, aka queue dodging. I don't believe the client gives this message if you try to close it after actually disconnecting from a game that you have been connected to? I seem to recall it usually says "Are you sure you want to log out?". This seems to support the theory that the game effectively disappears; it never advances past the champion select point in the eyes of the server.
