  In this paper, we study the regularity for viscosity solutions of locally
uniformly elliptic equations and obtain a series of interior pointwise
$C^{k,\alpha}$ ($k\geq 1$, $0<\alpha<1$) regularity with smallness assumptions
on the solution and the right-hand term. As applications, we obtain various
interior pointwise regularity for several classical elliptic equations, i.e.,
the prescribed mean curvature equation, the Monge-Amp\`{e}re equation, the
$k$-Hessian equations, the $k$-Hessian quotient equations and the Lagrangian
mean curvature equation. Moreover, the smallness assumptions are necessary in
most cases (Remark 2.6, Remark 3.5, Remark 4.7, Remark 5.4 and Remark 6.5).
