  The inspiral and merger of eccentric binaries leads to gravitational
waveforms distinct from those generated by circularly merging binaries.
Dynamical environments can assemble binaries with high eccentricity and peak
frequencies within the {\it LIGO} band. In this paper, we study binary-single
stellar scatterings occurring in dense stellar systems as a source of
eccentrically-inspiraling binaries. Many interactions between compact binaries
and single objects are characterized by chaotic resonances in which the
binary-single system undergoes many exchanges before reaching a final state.
During these chaotic resonances, a pair of objects has a non-negligible
probability of experiencing a very close passage. Significant orbital energy
and angular momentum are carried away from the system by gravitational wave
(GW) radiation in these close passages and in some cases this implies an
inspiral time shorter than the orbital period of the bound third body. We
derive the cross section for such dynamical inspiral outcomes through
analytical arguments and through numerical scattering experiments including GW
losses. We show that the cross section for dynamical inspirals grows with
increasing target binary semi-major axis, $a$, and that for equal-mass binaries
it scales as $a^{2/7}$. Thus, we expect wide target binaries to predominantly
contribute to the production of these relativistic outcomes. We estimate that
eccentric inspirals account for approximately one percent of dynamically
assembled non-eccentric merging binaries. While these events are rare, we show
that binary-single scatterings are a more effective formation channel than
single-single captures for the production of eccentrically-inspiraling
binaries, even given modest binary fractions.
