So just to confirm, was After Effects the last Adobe application you installed, or was it Acrobat?
Also, if you try to use Open With, click the Browse button, navigate to the Adobe After Effects CS6\Support FIles folder and select the AfterFX.exe executable, does that "Always use the selected program ..." checkbox enable at that point?
Fix your file associations on a system level. You are simply running out of context menu handlers and therefore AE never registers. In any Explorer Window use the options in the File --> Organize menu. Find the AEP fiel type and remove all its associations, then re-register it by right-clicking on teh file and using the Open With... method to define a new default program.
I'm trying to do this too but what does 'use the options in the File--Organize menu mean? I see the Organize drop down menu below the words File Edit View Tools and Help but if I click on the Organize drop down menu all I get our cut, copy, etc...Layout, Folder and search options, Delete, Rename and such. Where are the options in the File--Organize menu that you mentioned?
Ok simple easy idiot proof method. Go to
C:\Program Files\Adobe\Adobe After Effects CS5.5\Support Files
Rename AfterFX.exe to AfterFX.exe.org or anything will do to make it a non.exe file. Then click on any .aep file. Windoze will say can't find AE CS5.5 and it will then allow go to the dialog window where you can change it to open with AE CS6. Check the always open with this app box and you're done. Go back and rename the changed CS5.5 file to AfterFX.exe and you're back to normal with all future .aep files to open in CS6.
Same process with any version of AfterFX.exe Windoze may have locked you in to.