We consider the two-dimensional potential flow of \emph{standing gravity waves on an infinitely deep water}, with \emph{no surface tension at the free surface}. Linear theory gives \emph{infinitely many eigenmodes for any rational value of the dimensionless parameter} $\mu=gT^{2}/2\pi\ lambda$ ($T$ and $\lambda$ being time and space periods). We use a formulation of Dyachenko \emph{et al} leading to a nonlocal second order partial differential equation, for which the existence of infinitely many \ emph{approximate solutions} at any order is known. For proving the existence of solutions with a given asymptotic expansion, we use an appropriate version of the Nash-Moser implicit function theorem, where the major difficulty is to invert the linearized operator near a non zero point. We can show the \emph{existence of standing waves for an infinite sequence of values of} $\mu$\emph{ tending to any rational number}).