The value this references depends on circumstances. Any unscoped (i.e. not in an object) function call will have this = window (DOMWindow, if you prefer). Anything that is part of a prototype, or has been changed using apply or bind, will have this as that object ("itself", if you prefer).
So, to illustrate. When you are instantiating using the new keyword, your function automatically inherits this as itself. When you call an IETF within it, this within this IETF will point to the global scope.
If you would like to avoid this, instead of calling testMe literally, you can always do this:
var myFunction = function() {
    var testMe = function() {
        console.log(this);
    }
    testMe.bind(this);
}
Which will, in turn, have this within testMe use the object as it should.