The .push method can take multiple arguments. You can use the spread operator to pass all the elements of the second array as arguments to .push:

>>> a.push(...b)

If your browser does not support ECMAScript 6, you can use .apply instead:

>>> a.push.apply(a, b)

Or perhaps, if you think it's clearer:

>>> Array.prototype.push.apply(a,b)

Please note that all these solutions will fail with a stack overflow error if array b is too long (trouble starts at about 100,000 elements, depending on the browser).
If you cannot guarantee that b is short enough, you should use a standard loop-based technique described in the other answer.

If you want to modify the original array instead of returning a new array, use .push()...

array1.push.apply(array1, array2);
array1.push.apply(array1, array3);

I used .apply to push the individual members of arrays 2 and 3 at once.

or...

array1.push.apply(array1, array2.concat(array3));

To deal with large arrays, you can do this in batches.

for (var n = 0, to_add = array2.concat(array3); n < to_add.length; n+=300) {
    array1.push.apply(array1, to_add.slice(n, n+300));
}

If you do this a lot, create a method or function to handle it.

var push_apply = Function.apply.bind([].push);
var slice_call = Function.call.bind([].slice);

Object.defineProperty(Array.prototype, "pushArrayMembers", {
    value: function() {
        for (var i = 0; i < arguments.length; i++) {
            var to_add = arguments[i];
            for (var n = 0; n < to_add.length; n+=300) {
                push_apply(this, slice_call(to_add, n, n+300));
            }
        }
    }
});

and use it like this:

array1.pushArrayMembers(array2, array3);

var push_apply = Function.apply.bind([].push);
var slice_call = Function.call.bind([].slice);

Object.defineProperty(Array.prototype, "pushArrayMembers", {
    value: function() {
        for (var i = 0; i < arguments.length; i++) {
            var to_add = arguments[i];
            for (var n = 0; n < to_add.length; n+=300) {
                push_apply(this, slice_call(to_add, n, n+300));
            }
        }
    }
});

var array1 = ['a','b','c'];
var array2 = ['d','e','f'];
var array3 = ['g','h','i'];

array1.pushArrayMembers(array2, array3);

document.body.textContent = JSON.stringify(array1, null, 4);