Q: Why after installing and then re-installing El Capitan on an iMac, mid 2010, version 10.11.4, will  iTunes not open?

Hi KarenLizC, 

Welcome to Apple Support Communities. 

If I understand correctly, you've reinstalled OS X several times but there's still an issue with iTunes not starting as expected. Try ruling out an issue with iTunes plug-ins by holding the Option and Command keys while launching iTunes, as described here:

iTunes 12 for Mac: Keyboard shortcuts

                 

Open iTunes in “safe mode” (without external plug-ins)

Option-Command (while opening iTunes)

If iTunes launches in safe mode, try quitting iTunes and then removing any outdated plug-ins from the folders outlined below.

iTunes: May become unresponsive when connecting iPhone, iPad, or iPod touch - Apple Support

1. Test with iTunes plug-ins removed. 
    
   • Quit iTunes.
    
   • Navigate to the following folder, based on your operating system:
       Windows XP:
       C:\Documents and Settings\[username]\Application Data\Apple Computer\iTunes\iTunes Plug-ins
       Windows Vista or Windows 7:
       C:\Users\[username]\App Data\Roaming\Apple Computer\iTunes\iTunes Plug-ins
       Mac OS X: 
       
     • ~/Library/iTunes/iTunes Plug-ins (The "~" represents your Home folder)
       
     • /Library/iTunes/iTunes Plug-ins
      
      
    
   • If any files are in this folder, move them to the Desktop temporarily.
    
   • Open iTunes and test to see if the issue is resolved. If so, you may leave these iTunes plug-ins removed. If you wish to re-use these iTunes plug-ins, consult the vendor for the iTunes plug-ins to see if any updates are available.

If you don't see the Library folder, just hold the Option key as described here:

OS X El Capitan: Library folder

The Library folder is hidden. If you need to open it for some reason, click the desktop to make sure you’re in the Finder, hold down the Option key, then choose Go > Library.

If the issue persists when launching iTunes in safe mode, try using the general troubleshooting steps here to isolate and resolve the issue with iTunes. 

Mac OS X: How to troubleshoot a software issue - Apple Support

Cheers.