Let G be a connected real algebraic group. An unrefinable chain of G is a
chain of subgroups G=G0>G1>...>Gt=1 where each Gi is a maximal
connected real subgroup of Gi−1. The maximal (respectively, minimal)
length of such an unrefinable chain is called the length (respectively, depth)
of G. We give a precise formula for the length of G, which generalises
results of Burness, Liebeck and Shalev on complex algebraic groups and also on
compact Lie groups. If G is simple then we bound the depth of G above and
below, and in many cases we compute the exact value. In particular, the depth
of any simple G is at most 9