Just to flesh out Geoff's comment....
You can (more or less) find the length of a subgroup chain in $GL_n(q)$ by considering the length of a subgroup chain of a Borel subgroup. The length of this chain is easy to calculate - it is a function of $\log_pq$ and the prime factorization of $q-1$.
To make this comment precise, refer to this paper:
Solomon, Ron; Turull, Alexandre, Chains of subgroups in groups of Lie type. III. J. London Math. Soc. (2) 44 (1991), no. 3, 437–444.
The main result states:
Theorem A∗: Let $p$ be a prime. There exists a positive integer $F(p)$ such that whenever $G=G_r(k)$ is a finite quasisimple group of Lie type with $|k|=p^m$ and $m\geq F(p)$, then $l(G)=l(B)+r$, where $B$ is a Borel subgroup of $G$; moreover, every chain in $G$ of maximal length includes a maximal parabolic subgroup.
(Here we write $l(G)$ for the length of a subgroup chain in a group $G$.) Of course $GL_r(k)$ is not quasisimple, but $SL_r(k)$ is (unless $r$ and $k$ are very small), so the result gives the info you need.
The other two papers in this series are also worth a look - the second has an additional author, Gary Seitz.