The boundedness of the Hardy–Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. Moreover, we give some bounds of the norm of the Hardy–Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of Harmonic Analysis such as maximal and sharp maximal functions; Calderón–Zygmund singular integrals, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of 2π-periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces.

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces (X,d,μ) and (Y,ρ,ν) are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.