  The complete growth series of a finitely generated group is given by
$\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the
group semiring. We study the $\mathbb NG$-rationality and $\mathbb
NG$-algebraicity of such series. We show that having dead ends of arbitrarily
large depths is an obstruction to $\mathbb NG$-rationality. In the case of the
$3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete
series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also
used to show that complete growth series of higher Heisenberg groups are not
$\mathbb NG$-rational for specific generating sets. Using a more general
version of this obstruction, we prove that complete growth series of some
lamplighter groups are not $\mathbb NG$-rational either.
