Can an electrified wire ever be safe against bare skin?
First, what is the lowest amperage needed to light an LED? I think there are blue LEDs that can light up at 5 mA. 10 mA is where shocks begin to get painful.


This is roughly correct. However it's very easy to design a circuit that can deliver 5 mA to an LED but wouldn't be able to deliver more than a couple microamps to a human body, because of the much higher resistance of a body.


  in a circuit with a battery, is there any scenario where that low voltage of a battery could deliver a painful or harmful shock if something went wrong?


For a ~1.5 V battery, it's pretty difficult to detect them in my experience.

As a kid, we used to test 9 V batteries by touching the two terminals to our tongues. It gives a pretty noticeable tingle. It might be dangerous for somebody with a heart condition or an implanted medical device.


  could you power that LED with a bare copper wire touching your skin, such as wire jewelry, or would that circuit short on your skin or provide an electric shock?


If you are not "at risk" (i.e., heart condition or implanted medical device) this will probably work fine.

You would need to separate the "to" and "from" wires from each other so they don't short to each other. 


  I'm trying to figure out of it's possible to wear a bare copper wire around your neck or wrist that powers an LED, or does that wire need to be insulated?


FWIW, there are kinds of insulation that almost look like they're not there. Clear varnish or clear plastic. These might be more practical for a wearable application.

Also remember that copper can stain your skin in some circumstances. This is probably more likely if you sweat while wearing it. Of course this applies to any copper jewelry, and isn't related to whether you are using the copper in an electric circuit.