  Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra
$\mathfrak{h}$. Also fix a weight $\lambda\in\mathfrak{h}^*$. For
$M(\lambda)\twoheadrightarrow V$ an arbitrary highest weight
$\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula
for the weights of $V$. This is novel even in finite type, and is obtained from
$\lambda$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in
the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$.
  Our proofs use and reveal a finite family (for each $\lambda$) of "higher
order Verma modules" $\mathbb{M}(\lambda,\mathcal{H})$ - these are all of the
universal modules for weight-considerations. They (i) generalize and subsume
parabolic Verma modules $M(\lambda,J)$, and (ii) have pairwise distinct
weight-sets, which exhaust the weight-sets of all modules
$M(\lambda)\twoheadrightarrow V$. As an application, we explain the sense in
which the modules $M(\lambda)$ of Verma and $M(\lambda,J_V)$ of Lepowsky are
respectively the zeroth and first order upper-approximations of every $V$, and
continue to higher order upper-approximations
$\mathbb{M}_k(\lambda,\mathcal{H}_V)$ (and to lower-approximations). We also
determine the $k$th order integrability of $V$, for all $k\geq 0$.
  We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$,
which is a higher order parabolic analogue that contains the higher order Verma
modules $\mathbb{M}(\lambda,\mathcal{H})$. We show that
$\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study
of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over
$\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution
for the universal modules $\mathbb{M}(\lambda,\mathcal{H})$ in certain cases;
this yields a Weyl-type character formula for them, and involves the action of
a parabolic Weyl semigroup.
