= Solution
Use the standard defining properties of <Category O>: finite generation over $U(\mathfrak g)$, a <direct sum> of finite-dimensional <weight spaces>, and locally finite <positive root> action. In particular the weights lie in a finite union of downward <root> cones. Submodules and quotients retain the <representation weight> bounds and finite <weight spaces>; finite generation of submodules follows from the <Poincaré-Birkhoff-Witt theorem> and the noetherian property of $U(\mathfrak g)$.
Choose finitely many <weight vector> generators. The sum $T$ of their full <weight spaces> is finite-dimensional and preserved by the centre. The <kernel> of its central action has finite codimension in $Z(U(\mathfrak g))$ and kills the generators, hence, by centrality, all of $M$. Thus only finitely many central characters $\theta_1,\ldots,\theta_r$ can occur in any subquotient: they are the characters of a fixed finite-dimensional commutative <quotient> algebra of the centre.
By the <Harish-Chandra isomorphism> each $\theta_i$ determines one finite Weyl orbit of shifted weights. Let $S$ be the finite union of these orbits after subtracting $\rho$. Every nonzero subquotient has a nonzero highest-<weight vector>, using the bounded-above weights. Within its finite-dimensional <highest weight> space, the commuting central operators have a common <eigenvector>. Part (e) and the central-character restriction show that its <highest weight> belongs to $S$.
Define for any subquotient $N$
$$
d(N)=\sum_{\nu\in S}\dim N_\nu.
$$
It is finite, additive in short exact sequences, and positive whenever $N\ne0$. Hence each strict step of a <submodule> chain consumes at least one unit of $d(M)$. Refining a nonsimple factor into a proper <submodule> and <quotient> cannot continue more than $d(M)$ times. It ends with simple factors, proving
$$
\boxed{M\text{ has a composition series of finite length, at most }d(M).}
$$
The argument proves the assertion for every object of $\mathcal O$, rather than just for a cyclic <highest weight> object.
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