While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop around another loop , while both are linked to a third loop . We study this three-loop braiding in the context of gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with symmetry. We find that different short-range entangled bosonic states with the same symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.