  We prove large-scale $C^\infty$ regularity for solutions of nonlinear
elliptic equations with random coefficients, thereby obtaining a version of the
statement of Hilbert's 19th problem in the context of homogenization. The
analysis proceeds by iteratively improving three statements together: (i) the
regularity of the homogenized Lagrangian $\bar{L}$, (ii) the commutation of
higher-order linearization and homogenization, and (iii) large-scale
$C^{0,1}$-type regularity for higher-order linearization errors. We
consequently obtain a quantitative estimate on the scaling of linearization
errors, a Liouville-type theorem describing the polynomially-growing solutions
of the system of higher-order linearized equations, and an explicit
(heterogenous analogue of the) Taylor series for an arbitrary solution of the
nonlinear equations---with the remainder term optimally controlled. These
results give a complete generalization to the nonlinear setting of the
large-scale regularity theory in homogenization for linear elliptic equations.
