A possible solution...
As I had hoped, I was able to bring this non charging phone back to life.
So after having experienced this twice, I think I have a solution.
In my case the battery not accepting a charge seemed to be caused by draining the battery down, doing a partial charge, then using phone for a bit, then charging a bit more. I think this has to do with battery memory or something. In general batteries prefer to long cycle, fully charge and fully drain. I'm pretty sure this was caused by not doing that.
Before determining if this solution will work for your phone, do all the previously recommended steps. Including: testing your phone with a different cable/ charger, cleaning contacts, etc.
In my case the phone indicated it was charging, but battery status would non change, even after charging for several hours.
The solution seems to be very counterintuitive:
Let the the phone charge long enough to recieve a full charge (4hrs+?)
Note that the phone will indicate that it is charging but status will not change.
Disconnect the Phone from the charger.
Do a restart.
At this point things will look really grim.
You lose the phone functionality (super annoying) and get the must charge icon.
However your phone is fully charged it just doesn't know it,
Now you have to wait for 3-5 hours
This is where the patience comes in.
For some reason after this waiting period, with occasional attempts at starts/ restarts ...
a simple press of the start button and the phone comes back to life
and the battery is at 100%
I have had this experience twice.
And these seem to be the circumstances in which it died, and how it came back to life.
As mentioned in my previous post, The people at the Apple genius bar (Los Angeles Beverly Center) were completely clueless as to this problem or solution (I talked to 3 or 4 employees), and were quick to tell me that my phone was simply broken and they would do me the courtesy of selling me a replacement for $200.
Be patient and you can save yourself $200