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Stokes wave is a finite amplitude periodic gravity wave propagating withconstant velocity in inviscid fluid. Complex analytical structure of Stokeswave is analyzed using a conformal mapping of a free fluid surface of Stokeswave into the real line with fluid domain mapped into the lower complexhalf-plane. There is one square root branch point per spatial period of Stokeslocated in the upper complex half-plane at the distance $v_c$ from the realaxis. The increase of Stokes wave height results in approaching $v_c$ to zerowith the limiting Stokes wave formation at $v_c=0.$ The limiting Stokes wavehas $2/3$ power law singularity forming $2\pi/3$ radians angle on the crestwhich is qualitatively different from the square root singularity valid forarbitrary small but nonzero $v_c$ making the limit of zero $v_c$ highlynontrivial. That limit is addressed by crossing a branch cut of a square rootinto the second and subsequently higher sheets of Riemann surface to findcoupled square root singularities at the distances $\pm v_c$ from the real axisat each sheet. The number of sheets is infinite and the analytical continuationof Stokes wave into all these sheets is found together with the seriesexpansion in half-integer powers at singular points within each sheet. It isconjectured that non-limiting Stokes wave at the leading order consists of theinfinite number of nested square root singularities which also implies theexistence in the third and higher sheets of the additional square rootsingularities away from the real and imaginary axes. These nested square rootsform $2/3$ power law singularity of the limiting Stokes wave as $v_c$ vanishes.

Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface