One single account is having this issue: I open LaunchPad and login. The play button goes to playing almost instantly and sits there forever. The game doesn't actually launch as I can see only one EQ game (my other acct) in taskmanager. If I use another account I am able to hit play and the game launches correctly and all is well.
In the top right left of Logout in the drop-down, does it have the account name of the account you want to play or the one already loaded? If it's not the one you want, hit the drop-down arrow and select the one you want, or hit the Login to a New Account link to add that account to the list. Yther Ore.
If it's not as Yther describes, try reinstalling DirectX. I had a similar problem and only discovered the problem by trying to start eqgame.exe by itself and bypassing the launcher. I then got a message regarding a missing file which I tracked down to a faulty earlier DirectX installer I had used.
half a dozen log in attempts...finally get in...lag is too fierce to play...annual renewal coming up soon...thinking about not renewing...first time since 2004