How to edit the "ME" contact? On my Galaxy Nexus running (stock, non-rooted) ICS, there is a contact entry for "ME".

This is different from the contact entry farther down the list with my actual name on it.
And the contact for "ME" actually contains some incorrect information. Some non-existent email addresses, etc. I don't even know where these came from, as they aren't anywhere in my contacts lists nor in my G+ profile. 

When I try to edit "ME", it does not give me the option to modify the email addresses. 
And it also doesn't let me Join this contact with the other instance of myself.

How can I edit this "ME" contact to show correct information?
Or can I somehow delete it and choose a different contact to represent ME?
I also had the same problem and it took me 2 days to fix this

Try going to contacts.google.com.

Find your profile (should have your gmail on email address)

Re-add the faulty data (it was an old phone number in my case), save, and force your nexus to sync contacts

Now delete the faulty data, sava, and force sync again.

If that didnt work, do the same but with your google+ profile: re-add faulty data, save, and force sync, then delete it.

I did both of the above, so I'm not exactly sure which method worked

Edit: just remembered there was one other thing i did after all of the above:

Go to google plus app on your nexus

Click "settings" and uncheck the contacts syncing.

The old phone number was removed at this point. And turning sync back on did not make it reappear

Hope that helped!