For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoids.

For example take $C$ to be the poset with 6 elements ordered as a,b < c,d < e,f and when you localize at all arrows you get the $2$-sphere $\mathbb{S}^2$. So the simplicial localization will have all object isomorphic and having a simplicial set of endomorphisms for each object equivalent to $\Omega(\mathbb{S}^2)$.

For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoid.

For example take $C$ to be the poset with 6 elements ordered as a,b < c,d < e,f and when you localize at all arrows you get the $2$-sphere $\mathbb{S}^2$. So the simplicial localization will have all objects isomorphic and having a simplicial set of endomorphisms for each object equivalent to $\Omega(\mathbb{S}^2)$.