  Stokes wave is a finite amplitude periodic gravity wave propagating with
constant velocity in inviscid fluid. Complex analytical structure of Stokes
wave is analyzed using a conformal mapping of a free fluid surface of Stokes
wave into the real line with fluid domain mapped into the lower complex
half-plane. There is one square root branch point per spatial period of Stokes
located in the upper complex half-plane at the distance $v_c$ from the real
axis. The increase of Stokes wave height results in approaching $v_c$ to zero
with the limiting Stokes wave formation at $v_c=0.$ The limiting Stokes wave
has $2/3$ power law singularity forming $2\pi/3$ radians angle on the crest
which is qualitatively different from the square root singularity valid for
arbitrary small but nonzero $v_c$ making the limit of zero $v_c$ highly
nontrivial. That limit is addressed by crossing a branch cut of a square root
into the second and subsequently higher sheets of Riemann surface to find
coupled square root singularities at the distances $\pm v_c$ from the real axis
at each sheet. The number of sheets is infinite and the analytical continuation
of Stokes wave into all these sheets is found together with the series
expansion in half-integer powers at singular points within each sheet. It is
conjectured that non-limiting Stokes wave at the leading order consists of the
infinite number of nested square root singularities which also implies the
existence in the third and higher sheets of the additional square root
singularities away from the real and imaginary axes. These nested square roots
form $2/3$ power law singularity of the limiting Stokes wave as $v_c$ vanishes.
