How to edit the "ME" contact?
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!