  Let $G$ be a connected Lie group. An unrefinable chain of $G$ is a chain of
subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal
connected subgroup of $G_{i-1}$. In this paper, we introduce the notion of the
length (respectively, depth) of $G$, defined as the maximal (respectively,
minimal) length of such a chain, and we establish several new results for
compact groups. In particular, we compute the exact length and depth of every
compact simple Lie group, and draw conclusions for arbitrary connected compact
Lie groups $G$. We obtain best possible bounds on the length of $G$ in terms of
its dimension, and characterize the connected compact Lie groups that have
equal length and depth. The latter result generalizes a well known theorem of
Iwasawa for finite groups. More generally, we establish a best possible upper
bound on $\dim G'$ in terms of the chain difference of $G$, which is its length
minus its depth.
