  Black holes are thought to describe the geometry of massive, dark compact
objects in the universe. To further support and quantify this long-held belief
requires knowledge of possible, if exotic alternatives. Here, we wish to
understand how compact can self-gravitating solutions be. We discuss theories
with a well-posed initial value problem, consisting in either a single
self-interacting scalar, vector or both. We focus on spherically symmetric
solutions, investigating the influence of self-interacting potentials into the
compactness of the solutions, in particular those that allow for flat-spacetime
solutions. We are able to connect such stars to hairy black hole solutions,
which emerge as a zero-mass black hole. We show that such stars can have light
rings, but their compactness is never parametrically close to that of black
holes. The challenge of finding black hole mimickers to investigate full
numerical-relativity binary setups remains open.
