Perfect use case for recursion, which could handle even deeper structure:

function flatten(ary) {
    var ret = [];
    for(var i = 0; i < ary.length; i++) {
        if(Array.isArray(ary[i])) {
            ret = ret.concat(flatten(ary[i]));
        } else {
            ret.push(ary[i]);
        }
    }
    return ret;
}

flatten([[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]]) // [0, 1, 2, 3, 4, 5]

Alternatively, as an Array method:

Array.prototype.flatten = function() {
    var ret = [];
    for(var i = 0; i < this.length; i++) {
        if(Array.isArray(this[i])) {
            ret = ret.concat(this[i].flatten());
        } else {
            ret.push(this[i]);
        }
    }
    return ret;
};

[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flatten() // [0, 1, 2, 3, 4, 5]

EDIT #1: Well, think it a little bit functional way (except for the named recursion which should be using Y-combinator for pure functional :D).

function flatten(ary) {
  return ary.reduce(function(a, b) {
    if (Array.isArray(b)) {
      return a.concat(flatten(b))
    }
    return a.concat(b)
  }, [])
}

Let's adopt some ES6 syntax which makes it even shorter, in one line.

const flatten = (ary) => ary.reduce((a, b) => a.concat(Array.isArray(b) ? flatten(b) : b), [])

But remember, this one cannot be applied as an array method, because arrow functions don't have theirs own this.

EDIT #2: With the latest Array.prototype.flat proposal this is super easy. The array method accepts an optional parameter depth, which specifies how deep a nested array structure should be flattened (default to 1).

[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flat()  // [[[[0]], [1]], [[[2], [3]]], [[4], [5]]]
[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flat(2) // [[[0]], [1], [[2], [3]], [4], [5]]
[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flat(3) // [[0], 1, [2], [3], 4, 5]
[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flat(4) // [0, 1, 2, 3, 4, 5]

So to flatten an array of arbitrary depth, just call flat method with Infinity.

[[[[[0]], [1]], [[[2], [3]]], [[4], [5]]]].flat(Infinity) // [0, 1, 2, 3, 4, 5]