Is it possible to get the non-enumerable inherited property names of an object?

A cleaner solution using recursion:

function getAllPropertyNames (obj) {
    const proto     = Object.getPrototypeOf(obj);
    const inherited = (proto) ? getAllPropertyNames(proto) : [];
    return [...new Set(Object.getOwnPropertyNames(obj).concat(inherited))];
}

Edit

More generic functions:

function walkProtoChain (obj, callback) {
    const proto     = Object.getPrototypeOf(obj);
    const inherited = (proto) ? walkProtoChain(proto, callback) : [];
    return [...new Set(callback(obj).concat(inherited))];
}

function getOwnNonEnumPropertyNames (obj) {
    return Object.getOwnPropertyNames(obj)
        .filter(p => !obj.propertyIsEnumerable(p));
}

function getAllPropertyNames (obj) {
    return walkProtoChain(obj, Object.getOwnPropertyNames);
}

function getAllEnumPropertyNames (obj) {
    return walkProtoChain(obj, Object.keys);
}

function getAllNonEnumPropertyNames (obj) {
    return walkProtoChain(obj, getOwnNonEnumPropertyNames);
}

This same template can be applied using Object.getOwnPropertySymbols, etc.

Since getOwnPropertyNames can get you non-enumerable properties, you can use that and combine it with walking up the prototype chain.

function getAllProperties(obj){
    var allProps = []
      , curr = obj
    do{
        var props = Object.getOwnPropertyNames(curr)
        props.forEach(function(prop){
            if (allProps.indexOf(prop) === -1)
                allProps.push(prop)
        })
    }while(curr = Object.getPrototypeOf(curr))
    return allProps
}

I tested that on Safari 5.1 and got

> getAllProperties([1,2,3])
["0", "1", "2", "length", "constructor", "push", "slice", "indexOf", "sort", "splice", "concat", "pop", "unshift", "shift", "join", "toString", "forEach", "reduceRight", "toLocaleString", "some", "map", "lastIndexOf", "reduce", "filter", "reverse", "every", "hasOwnProperty", "isPrototypeOf", "valueOf", "__defineGetter__", "__defineSetter__", "__lookupGetter__", "propertyIsEnumerable", "__lookupSetter__"]

---

Update: Refactored the code a bit (added spaces, and curly braces, and improved the function name):

function getAllPropertyNames( obj ) {
    var props = [];

    do {
        Object.getOwnPropertyNames( obj ).forEach(function ( prop ) {
            if ( props.indexOf( prop ) === -1 ) {
                props.push( prop );
            }
        });
    } while ( obj = Object.getPrototypeOf( obj ) );

    return props;
}