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We consider the Hamiltonian formulation of Horava gravity in arbitrary dimensions, which has been proposed as a renormalizable gravity model for quantum gravity without the ghost problem. We study the "full" constraint analysis of the "non-projectable" Horava gravity whose potential, V(R), is an arbitrary function of the (intrinsic) Ricci scalar R. We find that there exist generally distinct cases of this theory, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints (Cases A, C) or just fixes the associated Lagrange multipliers (Case B), or (ii) whether the IR Lorentz-deformation parameter λis at the conformal point (Case C) or not (Cases A, B). It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is "one additional phase-space degree of freedom", representing an extra (odd) scalar graviton mode. This would resolve the long-standing debates about the extra graviton modes and achieves the dynamical consistency of the Horava gravity, at the "fully non-linear" level. Several exact solutions are also considered as some explicit examples of the new constraints. The structure of the newly obtained, "extended" constraint algebra seems to be generic to Horava gravity and its general proof would be a challenging problem. Some other challenging problems, which include the path integral quantization and the Dirac bracket quantization are discussed also.