Is there a way to get the name of the caller function within the callee?

You used to be able to do arguments.caller.name, but this is deprecated in Javascript 1.3.

arguments.callee.caller.name (or just showMe.caller.name) is another way to go. This is non-standard, and not supported in strict mode, but otherwise currently supported in all major browsers (ref).

Try callee.caller like this

 function showMe() {
        // should log the runMe as the caller and showMe as callee
        console.log('Callee: ',arguments.callee.name)
        console.log('Caller: ',arguments.callee.caller.name);
      }