  A remarkable feature of chaos in many-body quantum systems is the existence
of a bound on the quantum Lyapunov exponent. An important question is to
understand what is special about maximally chaotic systems which saturate this
bound. Here we provide further evidence for the `hydrodynamic' origin of chaos
in such systems, and discuss hallmarks of maximally chaotic systems. We first
provide evidence that a hydrodynamic effective field theory of chaos we
previously proposed should be understood as a theory of maximally chaotic
systems. We then emphasize and make explicit a signature of maximal chaos which
was only implicit in prior literature, namely the suppression of exponential
growth in commutator squares of generic few-body operators. We provide a
general argument for this suppression within our chaos effective field theory,
and illustrate it using SYK models and holographic systems. We speculate that
this suppression indicates that the nature of operator scrambling in maximally
chaotic systems is fundamentally different to scrambling in non-maximally
chaotic systems. We also discuss a simplest scenario for the existence of a
maximally chaotic regime at sufficiently large distances even for non-maximally
chaotic systems.
