A "non-trivial" example of a Cauchy sequence that does not converge?
For any metric space $Q$ we can define the completion, that is a (bigger) metric space $R$ such that $Q$ is a (dense) subspace of $R$ and all Cauchy sequences in $Q$ have a limit in $R$. So the Cauchy sequences in $Q$ "do converge but what they converge to is not in the space". This is precisely one way of defining $\mathbb R$ from $\mathbb Q$.
In fact, there are no examples at all ("trivial" or otherwise). Any metric space admits a completion, and every Cauchy sequence in the original space is again Cauchy in the completion, where it converges.