  The transitivity degree of a group $G$ is the supremum of all integers $k$
such that $G$ admits a faithful $k$-transitive action. Few obstructions are
known to impose an upper bound on the transitivity degree for infinite groups.
The results of this article provide two new classes of groups whose
transitivity degree can be computed, as a corollary of a classification of all
$3$-transitive actions of these groups. More precisely, suppose that $G$ is a
subgroup of the homeomorphism group of the circle
$\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree
$\mathsf{Aut}(\mathbb{T})$. Under natural assumptions on the stabilizers of the
action of $G$ on $\mathbb{S}^1$ or $\partial \mathbb{T}$, we use the dynamics
of this action to show that every faithful action of $G$ on a set that is at
least $3$-transitive must be conjugate to the action of $G$ on one of its
orbits in $\mathbb{S}^1$ or $\partial \mathbb{T}$.
