  We develop an optimal regularity theory for $L^p$-viscosity solutions of
fully nonlinear uniformly elliptic equations in nondivergence form whose
gradient growth is described through a Hamiltonian function with measurable and
possibly unbounded coefficients. Our approach treats both superlinear and
sublinear gradient regimes in a unified way.
  We show $C^{0,\alpha}$, $C^{0,\textrm{Log-Lip}}$, $C^{1,\alpha}$, $
C^{1,\textrm{Log-Lip}}$ and $C^{2,\alpha}$ regularity estimates, by displaying
the growth allowed to the Hamiltonian in order to deal with an unbounded
nonlinear gradient coefficient, whose integrability in turn gets worse as we
approach the quadratic regime.
  Moreover, we find proper compatibility conditions for which our regularity
results depend intrinsically on the integrability of the underlying source
term. As a byproduct of our methods, we prove a priori BMO estimates; sharp
regularity to associated recession and flat profiles under relaxed convexity
assumptions; improved regularity for a class of singular PDEs; and a Perron
type result under unbounded ingredients.
