  Maximal green sequences are particular sequences of mutations of quivers
which were introduced by Keller in the context of quantum dilogarithm
identities and independently by Cecotti-C\'ordova-Vafa in the context of
supersymmetric gauge theory. The existence of maximal green sequences for
exceptional finite mutation type quivers has been shown by
Alim-Cecotti-C\'ordova-Espahbodi-Rastogi-Vafa except for the quiver $X_7$. In
this paper we show that the quiver $X_7$ does not have any maximal green
sequences. We also generalize the idea of the proof to give sufficient
conditions for the non-existence of maximal green sequences for an arbitrary
quiver.
