  In maximally chaotic quantum systems, a class of out-of-time-order
correlators (OTOCs) saturate the Maldacena-Shenker-Stanford (MSS) bound on
chaos. Recently, it has been shown that the same OTOCs must also obey an
infinite set of (subleading) constraints in any thermal quantum system with a
large number of degrees of freedom. In this paper, we find a unique analytic
extension of the maximally chaotic OTOC that saturates all the subleading chaos
bounds which allow saturation. This extremally chaotic OTOC has the feature
that information of the initial perturbation is recovered at very late times.
Furthermore, we argue that the extremally chaotic OTOC provides a
K\"{a}llen-Lehmann-type representation for all OTOCs. This representation
enables the identification of all analytic completions of maximal chaos as
small deformations of extremal chaos in a precise way.
