Recall that, for probability measures $p$ and $q$ on a finite set $X$, the Kullback-Leibler divergence $$ D(p||q) = \sum_{x\in X} p(x) \log \dfrac{p(x)}{q(x)} $$ is famously not a metric, in particular it does not satisfy a triangle inequality.
However, in principle one could look at the following quantity $$ d(p,q) := \inf_{n\in \Bbb{N}} \inf_{p_1,\dots,p_n} \Big( d(p,p_1) + d(p_1,p_2) + \dots + d(p_n, q) \Big) , $$ which does satisfy the triangle inequality, it is the largest quasimetric which is less or equal than $D(p||q)$.
Has this quantity been studied anywhere? Is it trivial (i.e. zero)?