  A partial regularity theorem is presented for minimisers of $k$th-order
functionals subject to a quasiconvexity and general growth condition. We will
assume a natural growth condition governed by an $N$-function satisfying the
$\Delta_2$ and $\nabla_2$ conditions, assuming no quantitative estimates on the
second derivative of the integrand; this is new even in the $k = 1$ case. These
results will also be extended to the case of strong local minimisers.
