We introduce the notion of the depth of a finite group G, defined as the
minimal length of an unrefinable chain of subgroups from G to the trivial subgroup. In
this paper we investigate the depth of (non-abelian) finite simple groups. We determine
the simple groups of minimal depth, and show, somewhat surprisingly, that alternating
groups have bounded depth. We also establish general upper bounds on the depth of
simple groups of Lie type, and study the relation between the depth and the much studied
notion of the length of simple groups. The proofs of our main theorems depend (among
other tools) on a deep number-theoretic result, namely, Helfgott’s recent solution of the
ternary Goldbach conjecture.