  While winding a particle-like excitation around a loop-like excitation yields
the celebrated Aharonov-Bohm phase, we find a distinctive braiding phase in the
absence of such mutual winding. In this work, we propose an exotic
particle-loop-loop braiding process, dubbed the \emph{Borromean-Rings
braiding}. In the process, a particle moves around two unlinked loops, such
that its trajectory and the two loops form the Borromean-Rings or more general
Brunnian links. As the particle trajectory does not wind with any of the loops,
the resulting braiding phase is fundamentally different from the Aharonov-Bohm
phase. We derive an explicit expression for the braiding phase in terms of the
underlying Milnor's triple linking number. We also propose Topological Quantum
Field Theories consisting of an $AAB$-type topological term which realize the
braiding statistics.
