Ryan's answer generalizes. I prefer $\iota_1, \iota_2$ for the inclusions of wedge summands, and then $\omega_1, \omega_2$ for forms which generate cohomology supported on each of the wedge summands. Then I claim that a homotopy invariant which evaluates non-trivially on $[[\cdots[\iota_1,\iota_2],\iota_2],\cdots],\iota_2]$ is
$f \mapsto \int_{S^n} d^{-1} \left( \cdots d^{-1} \left( d^{-1} \left(d^{-1} f^*(\omega_1) \wedge f^*(\omega_2)) \wedge f^*(\omega_2) \right) \wedge f^*(\omega_2) \right) \cdots \right) \wedge f^*(\omega_2),$ where each $d^{-1}$ denotes a choice some form to cobound what is in each case a closed and thus exact form of degree less than $n$ on $S^n$. Since there is such a Whitehead product in every degree greater than or equal to two, this claim answers the question.
The proof that this integral is homotopy invariant is elementary, along the lines which Ryan gives in the example of his answer. There is also probably a more direct "hands-on" proof that this integral evaluates to one on the Whitehead product given, but I am going to cite my work with Ben Walter, namely https://arxiv.org/abs/0809.5084 (not the paper Ryan linked to, though I appreciate his seeing the relevance of our ideas). This integral is a case of a Hopf invariant, which is the main object of study of the paper. This integral is the Hopf invariant associated to the element of the bar construction $|\omega_1|\omega_2| \omega_2| \cdots |\omega_2|$. To see it evaluates as claimed I prefer to use the bracket-cobracket compatibility theorem, namely Theorem 1.12 of the paper, from which this is quick inductive combinatorics. While the present MO answer is thus not self-contained, note that Theorem 1.12 occurs just on page 7, and I would call its proof "hands-on".
The cited paper was written to solve exactly these types of questions. We give an algorithm called weight reduction which can be used to associate integral(s - non-unique) to any cycle in the (Lie coalgebraic) bar construction, which are homotopy periods. This association is an isomorphism between the homology of this bar construction and the linear dual of homotopy. Though Sullivan in his seminal paper pointed to how one can produce complete rational homotopy periods (I wish we could call this "rational cohomotopy") through minimal models, what Ben and I find is that the Quillen functor approach is, perhaps surprisingly, more readily tied to geometry. The general principle which follows from our work is that one can detect all rational homotopy through numerical invariants which are "linking with correction," governed by the Lie coalgebraic bar construction. Thus, geometry, combinatorics, algebra and functorial formalism all work together as well as one could hope for in giving a sharp resolution to the homotopy periods question in the simply connected setting. (The non-simply connected case is ongoing research that I am having PhD students work on.) And as Ryan says, I have recorded lectures on this subject. The best place to see a summary of and links to the lectures is here https://pages.uoregon.edu/dps/GeometricAlgebraicTopology/ The first three lectures cover these Hopf invariants.