  This paper introduces new mixed formulations and discretizations for
$m$th-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary
$m=1,2,3,\dots$ based on novel Helmholtz-type decompositions for tensor-valued
functions. The new discretizations allow for ansatz spaces of arbitrary
polynomial degree and the lowest-order choice coincides with the non-conforming
FEMs of Crouzeix and Raviart for $m=1$ and of Morley for $m=2$. Since the
derivatives are directly approximated, the lowest-order discretizations consist
of piecewise affine and piecewise constant functions for any $m=1,2,\dots$
Moreover, a uniform implementation for arbitrary $m$ is possible. Besides the a
priori and a posteriori analysis, this paper proves optimal convergence rates
for adaptive algorithms for the new discretizations.
