I am trying to explain metric spaces and finding it hard to come up with simple to understand, interesting examples of metrics that are not translation invariant. The audience is people who are jsut now studying general metric spaces.
I just looked up "metric spaces" on wikipedia and two examples stood out:
1) Give the positive real line the metric d(x,y) = |log(x/y)|
2) In Euclidean space suppose that instead of considering the direct distance from x to y, we want to travel via 0, then the distance is given by:
d(x,y) = |x| + |y|
unless x=y in which case d(x,x) = 0.
I couldn't say where these appear naturally in pure mathematics, but they seem easy enough to understand and it is certainly not unthinkable that such space may at the very least be used in applied mathematics and therefore that our abstract study of metric spaces should include them.
You should of course also keep in mind that lots of metrics are on spaces without a vector space structure so the term "translation invariant" does not even make sense for general metric spaces.