  We provide new examples of groups without rational cross-sections (also
called regular normal forms), using connections with bounded generation and
rational orders on groups. Specifically, our examples are extensions of
infinite torsion groups, groups of Grigorchuk type, wreath products similar to
$C_2\wr(C_2\wr \mathbb Z)$ and $\mathbb Z\wr F_2$, a group of permutations of
$\mathbb Z$, and a finitely presented HNN extension of the first Grigorchuk
group. This last group is the first example of finitely presented group with
solvable word problem and without rational cross-sections. It is also not
autostackable, and has no left-regular complete rewriting system.
