Results of [Culler and Shalen][1] together with tameness, density, etc. imply that there exists a number $V$ such that if $M$ is a hyperbolic 3-manifold of volume $>V$, then the Margulis constant of $M$ is $\geq \log(3)$. This indicates that one ought to be able to compute the Margulis constant, making it an a priori trivial problem. To prove this, you take a sequence of 2-generator groups realizing the Margulis constant for manifolds with volume approaching $\infty$. In the limit, the Margulis constant is $>\log(3)$ by Culler-Shalen, so one concludes that there is some bound on volume for manifolds with Margulis constant $<\log(3)$. 

**Addendum:** Recently [Yarmola (in joint work with Futer and Gabai) has announced][2] that they can show that the Margulis constant $\mu_3 > .5$. 


  [1]: http://www.ams.org/mathscinet-getitem?mr=1135928
  [2]: https://math.yale.edu/event/towards-margulis-constant-hyperbolic-3-manifolds