I have to confess to a dislike of the word "interesting" for these constructions! So far, all that's been said is:
Any metric is equivalent to a bounded metric, so "boundedness" is not a topological property.
Countable products of metric spaces are metrisable, so the category of metric spaces has countable products. (Incidentally this construction fails for arbitrary small products.)
To that I can add arbitrary small coproducts: if $x,y$ are in the same component, take $d(x,y) = min(d_X(x,y),1)$ and if $x,y$ are in different components, take $d(x,y) = 1$.
What this shows is that from a topological point of view, it's usually best to have a bounded metric.
But I would say that an interesting construction is one that is not functorial, for example, something where you replace the given metric locally by another one. For example, on a manifold we can always replace a given metric by one which is flat near a point (indeed, flat near quite a large number of points) but we can't necessarily replace it by one which is flat everywhere.