It is assumed the solitons could propagate only on the surface of finite depth fluid. We show numerically that the strong localized perturbation of free fluid surface could propagate on the surface of deep fluid also. They are not solitons in a "classical" sense of this term; they are "breathers". It means that the motion of surface is a periodic function on time in a certain moving framework. This framework moves with the group velocity. Breathers of small steepness(ka < 0.07) are known for years as solitonic solutions of the Nonlinear Schrodinger Equation (NLSE) (Do not confuse with the "breather" solution of the Nonlinear Schrodinger Equation). Our new result is following: The breathers exits without decay up to very high steepness (at least to ka=0.5). These steep giant breathers can be identified with Freak Waves. The most plausible explanation of the giant breather stability is integrability of the Euler equation describing a potential flow of deep ideal fluid with free surface. So far we don't have a direct proof of this extremely strong statement, but we have a whole string of indirect evidences of this integrability.