  We study the limiting behavior of large-amplitude standing waves on deep
water using high-resolution numerical simulations in double and quadruple
precision. While periodic traveling waves approach Stokes's sharply crested
extreme wave in an asymptotically self-similar manner, we find that standing
waves behave differently. Instead of sharpening to a corner or cusp as
previously conjectured, the crest tip develops a variety of oscillatory
structures. This causes the bifurcation curve that parametrizes these waves to
fragment into disjoint branches corresponding to the different oscillation
patterns that occur. In many cases, a vertical jet of fluid pushes these
structures upward, leading to wave profiles commonly seen in wave tank
experiments. Thus, we observe a rich array of dynamic behavior at small length
scales in a regime previously thought to be self-similar.
