  In this investigation we study extreme vortex states defined as
incompressible velocity fields with prescribed enstrophy $\mathcal{E}_0$ which
maximize the instantaneous rate of growth of enstrophy $d\mathcal{E}/dt$. We
provide {an analytic} characterization of these extreme vortex states in the
limit of vanishing enstrophy $\mathcal{E}_0$ and, in particular, show that the
Taylor-Green vortex is in fact a local maximizer of $d\mathcal{E} / dt$ {in
this limit}. For finite values of enstrophy, the extreme vortex states are
computed numerically by solving a constrained variational optimization problem
using a suitable gradient method. In combination with a continuation approach,
this allows us to construct an entire family of maximizing vortex states
parameterized by their enstrophy. We also confirm the findings of the seminal
study by Lu & Doering (2008) that these extreme vortex states saturate (up to a
numerical prefactor) the fundamental bound $d\mathcal{E} / dt < C \,
\mathcal{E}^3$, for some constant $C > 0$. The time evolution corresponding to
these extreme vortex states leads to a larger growth of enstrophy than the
growth achieved by any of the commonly used initial conditions with the same
enstrophy $\mathcal{E}_0$. However, based on several different diagnostics,
there is no evidence of any tendency towards singularity formation in finite
time. Finally, we discuss possible physical reasons why the initially large
growth of enstrophy is not sustained for longer times.
