  A topological group $G$ is said to have a local $\omega^\omega$-base if the
neighbourhood system at identity admits a monotone cofinal map from the
directed set $\omega^\omega$. In particular, every metrizable group is such,
but the class of groups with a local $\omega^\omega$-base is significantly
wider. The aim of this article is to better understand the boundaries of this
class, by presenting new examples and counter-examples. Ultraproducts and
non-arichimedean ordered fields lead to natural families of non-metrizable
groups with a local $\omega^\omega$-base which nevertheless are Baire
topological spaces.
  More examples come from such constructions as the free topological group
$F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more
generally uniform) space $X$, as well as the free product of topological
groups. We show that 1) the free product of countably many separable
topological groups with a local $\omega^\omega$-base admits a local
$\omega^\omega$-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a
local $\omega^\omega$-base if and only if the finest uniformity of $X$ has a
$\omega^\omega$-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a
local $\omega^\omega$-base provided $X$ is separable and the finest uniformity
of $X$ has a $\omega^\omega$-base.
