The icon appears at the bottom of the screen as if it is going to start but doesn't. The icon will stay at the bottom of the screen but nothing else happens. Someone please help.
Here's a few trouble shooting steps: http://answers.ea.com/t5/The-Sims-3/Links-to-Mac-help-threads-and-guides/m-p/749664#U749664
If those don't help, I would suggest posting in the mac forum at thesims3.com (make sure your language is set to US): http://forum.thesims3.com/jforum/forums/show/205.page since macs are not well covered on the AnswerHQ.
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So the Launcher never opens after you try to start the game? It can take quite a long time for the Launcher to appear.
If you have waited for ages and it still hasn't opened then try going into Documents, finding your The Sims 3 folder and dragging the whole folder onto the desktop (you can keep it there so you can always retrieve downloads, saves and other important files from it later). Now try to launch again. The game will generate a fresh The Sims 3 folder in Documents.
If the game now works then you can deduce that there was something in the original folder that was causing problems. You can therefore move back some files (downloads, saved games etc) but be selective or you may end up in the same position as before. If you have CC and mods, make sure you only put back up-to-date versions.