  The set $A$ is an asymptotic nonbasis of order $h$ for an additive abelian
group $X$ if there are infinitely many elements of $X$ not in the $h$-fold
sumset $hA$. For all $h \geq 2$, this paper constructs new classes of
asymptotic nonbases of order $h$ for $\mathbf{Z}$ and for $\mathbf{N}_0$ that
are not subsets of maximal asymptotic nonbases.
