Let $M(\lambda)$ be the Verma module with highest weight $\lambda$. It is well-known that $M(\lambda)$ has a unique maximal submodule such that $\lambda$ is not a weight of it. Let $N$ be the unique maximal submodule of $M(\lambda)$. It is well-known that $M(\lambda)/N\cong L(\lambda)$, where $L(\lambda)$ is the unique simple quotient of $M(\lambda)$.
I would like to know whether $N$ has any maximal submodule.
If yes, let $N'$ be one of the maximal submodule of $N$.
Will $N/N'\cong L(\mu)$ for some $\mu\in\mathfrak{h}^*$?
The answer, I believe, is yes. However, a resolution of Verma modules was constructed by Bernstein-Gelfand-Gelfand. A description of the so-called BGG resolution, and a reference to the relevant paper, can be found on this Wikipedia page (Reference 11 in "Notes")