Strict $\infty$-groupoids are equivalent to crossed complexes, see paper available here, and the latter are often more convenient to handle, because of their analogies to chain complexes. The classifying space $BC$ of a crossed complex $C$ fibres over a $K(G,1)$ with fibre the classifying space of a chain complex, so up to homotopy a product of Eilenberg Mac Lane spaces. However crossed complexes are useful for discussing operations of the fundamental group, and in this respect have better realisation properties than chain complexes with operators, as JHC Whitehead observed in 1949.
The category of crossed complexes is monoidal closed so one can consider monoid objects in this category, and these contain some quadratic information.
However higher Whitehead products are well modelled by cat$^n$-groups ($n$-fold grouoids in the category of groups), and the equivalent category of crossed $n$-cubes of groups, see
Ellis, G. J. and Steiner, R., Higher-dimensional crossed modules and the homotopy groups
of $(n+1)$-ads, J. Pure Appl. Algebra, 46 (1987) 117--136.
and the modelling of weak pointed homotopy $(n+1)$-types by cat$^n$-groups was shown by Loday (JPAA 1982).
It is possible that quadratic homotopy information is modelled by double crossed complexes, etc.
Edit: I would like to add that Thierry Coquand has a cubical set model for type theory available from here.
December 12, 2015
I should add that in the usual homotopy theory there is a theory of strict higher homotopy groupoids with useful applications, see the 2011 book Nonabelian Algebraic Topology, but a homotopically defined functor with values in strict cubical groupoids is defined there only on filtered spaces. There are several presentations (Paris (workshop on HTT), Galway, Aveiro, ...) on my preprint page giving more background to this. A naive question is whether filtered topoi or something analogous could be useful in HTT? Just as many useful spaces have some kind of dimensionwise structure, so this also may be true for types, especially if they are useful for describing mathematical structures, as would be my intuition, which is limited!
The presentation at CT2015 Aveiro is entitled "A philosophy of modelling and computing homotopy types", and shows briefly how you get some specific nonabelian computations.