For an infinite-type surface $S$ with complete metric $d$ and compact exhaustion $\{K_n\}_{n=1}^\infty$, Vlamis, in "Notes on the Topology of Mapping Class Groups" Appendix A, defines a metric, $\rho$, on Homeo$(S)$ as follows:
for each $n$, define $\delta_n(f,g)=\min\{2^{-n},\sup\{d(f(x),g(x)):x\in K_n\}\}$, then set $\rho(f,g)=\sum_{n=1}^\infty \delta_n(f,g)$
By Arens' "A Topology for Spaces of Transformations", this metric coincides with the compact-open topology on Homeo$(S)$. Vlamis then goes on to define a new complete metric because he says $\rho$ is not complete, hinting that "it is critical that the sequence $\{f_n^{-1}\}$ is Cauchy whenever $\{f_n\}$ is Cauchy".
I see why this is necessary, but I'm having trouble coming up with an example where $\{f_n\}$ is Cauchy but $\{f_n^{-1}\}$ isn't, or in general showing that $\rho$ is necessarily not complete.
First, notice that the metric $\rho$ can be extended into a metric $\rho'$ on the space $\operatorname C(S,S)$ of all continuous functions from $S$ to $S$ by simply using the same formulae which were used to define it on $\operatorname{Homeo}(S)$, but with general continuous functions from $S$ to $S$ instead of just homeomorphisms.
It is easy to check that the topology on $\operatorname C(S,S)$ defined by $\rho'$ is the compact-open topology and that the metric $\rho'$ is complete. Hence, deciding whether or not a sequence of elements of $\operatorname{Homeo}(S)$ is Cauchy for $\rho$ is equivalent to deciding whether or not it is convergent in $\operatorname C(S,S)$ for the compact-open topology. This makes finding examples easier.
Consider for example $S=\mathbb R^2$, with $d=$ any complete metric compatible with the topology and $\lbrace K_n\rbrace$ any compact exhaustion, and for every $n\geq 1$, let $f_n=2^{-n}\operatorname{id}_{\mathbb R^2}$. Each $f_n$ is a homemorphism of $\mathbb R^2$ and $f_n^{-1}=2^n\operatorname{id}_{\mathbb R^2}$. So clearly, $\lbrace f_n\rbrace$ converges to $0$ in $\operatorname C(S,S)$ for the compact-open topology, and $\lbrace f_n^{-1}\rbrace$ is not convergent in $\operatorname C(S,S)$ for the compact-open topology.