  Let $A_N$ denote nondeterministic automatic complexity and \[
  L_{k,c}=\{x\in [k]^* : A_N(x)> |x|/c\}. \] In particular, $L_{k,2}$ is the
language of all $k$-ary words for which $A_N$ is maximal, while $L_{k,3}$ gives
a rough dividing line between complex and simple. Let $\mathbf{CFL}$ denote the
complexity class consisting of all context-free languages. While it is not
known that $L_{2,2}$ is infinite, Kjos-Hanssen (2017) showed that $L_{3,2}$ is
$\mathbf{CFL}$-immune but not $\mathbf{coCFL}$-immune. We complete the picture
by showing that $L_{3,2}\not\in\mathbf{coCFL}$.
  Turning to Boolean circuit complexity, we show that $L_{2,3}$ is
$\mathbf{SAC}^0$-immune and $\mathbf{SAC}^0$-coimmune. Here $\mathbf{SAC}^0$
denotes the complexity class consisting of all languages computed by
(non-uniform) constant-depth circuits with semi-unbounded fanin.
  As for arithmetic circuits, we show that
$\{x:A_N(x)>1\}\not\in\oplus\mathbf{SAC}^0$. In particular,
$\mathbf{SAC}^0\not\subseteq\oplus \mathbf{SAC}^0$, which resolves an open
implication from the Complexity Zoo.
