  Galileon models are a class of effective field theories that have recently
received much attention. They arise in the decoupling limit of theories of
massive gravity, and in some cases they have been treated in their own right as
scalar field theories with a specific nonlinearly realized global symmetry
(Galilean transformation). It is well known that in the presence of a source,
these Galileon theories admit superluminal propagating solutions, implying that
as quantum field theories they must admit a different notion of causality than
standard local Lorentz invariant theories. We show that in these theories it is
easy to construct closed timelike curves (CTCs) within the {\it naive} regime
of validity of the effective field theory. However, on closer inspection we see
that the CTCs could never arise since the Galileon inevitably becomes
infinitely strongly coupled at the onset of the formation of a CTC. This
implies an infinite amount of backreaction, first on the background for the
Galileon field, signaling the break down of the effective field theory, and
subsequently on the spacetime geometry, forbidding the formation of the CTC.
Furthermore the background solution required to create CTCs becomes unstable
with an arbitrarily fast decay time. Thus Galileon theories satisfy a direct
analogue of Hawking's chronology protection conjecture.
