  We show that the string worldsheet theory of Gaiotto-Maldacena holographic
duals to N=2 superconformal field theories generically fails to be classically
integrable. We demonstrate numerically that the dynamics of a winding string
configuration possesses a non-vanishing Lyapunov exponent. Furthermore an
analytic study of the Normal Variational Equation fails to yield a Liouvillian
solution. An exception to the generic non-integrability of such backgrounds is
provided by the non-Abelian T-dual of $AdS_5 \times S^5$. Here by virtue of the
canonical transformation nature of the T-duality classical integrability is
known to be present.
