Eccentric binary black holes Comparable-mass binary black hole inspirals and mergers are expected to be an important source of gravitational wave signals for current and future ground-based detectors. 
It is generally expected that such mergers occur as the end product of the evolutions of certain supergiant stellar binaries. In that case, since gravitational radiation strongly damps eccentricity, the inpiral should be "quasicircular" (circular over the orbital timescale, which is much shorter than the inspiral timescale).
This is unfortunate because the interesting relativity effects typically manifest only at substantial eccentricities. Therefore my question, which represents a certain amount of wishful thinking: are there any astrophysically-plausible mechanisms that might yield an inspiral with substantial eccentricity?
 A: Short answer: No.  There are numerous effects which can enhance the eccentricity of BH binaries.  This is a very active area of study (and thus fairly uncertain), but the leading mechanisms are tangential-deceleration preferentially at apocenter (by, e.g. dynamical friction or [perhaps] stellar scattering in massive BH systems) or excitation from a perturber (almost exclusively a stellar-mass BH-binary, with a massive tertiary).  The problem is that these effects act slowly, on secular timescales, while the Gravitational-Wave (GW) damping of eccentricity is extremely effective as the system nears inspiral --- and acts rapidly, on a dynamical timescale.
To observe the effects of eccentricity during merger, you would basically need to excite a near-unity eccentricity in only the final orbit --- which is extremely difficult to do (cough, effectively impossible, cough).
Slight Aside:
Just to be an especial-downer, let me point out that this doesn't really matter because high-eccentricity effects are practically unobservable.  This is because LIGO detections are based on templated searches which rely on observing hundreds to thousands of oscillations to make a statistical detection.  To observe high-eccentricity effects at the time of inspiral would require really high signal to noise.  We can make a quick estimate by noting that the Signal-to-Noise Ratio (SNR) is proportional to $1/r$ where $r$ is the distance to the binary (note this is not $1/r^2$).  A binary blackhole can be detected at an SNR of about 8 in each of 3 detectors, at about 1 Gpc.  Lets say that a factor of 10 higher SNR would yield observable eccentric effects (I think it's probably a lot worse than this...), which would require a 10x closer system, sampling a $10^3$ smaller volume of space, and thus a $10^3$ lower detection rate.  Let's say the first binary BH detection occurs after a month of operation (and we haven't heard yet...), this would suggest it would take another 100 yrs before a sufficiently nearby system were observed.  If we're lucky, LISA might be up within roughly that time --- which would have much better chances!
A: I actually wrote a paper on this. The prime motivation was to propose a unmodeled search as matched-filtering search aiming circular binaries will miss eccentric binaries. The argument regarding LIGO seeing an event in one month will change as the sensitivity improves. By the time LIGO gets to advanced sensitivity we will probably see around 10-40(roughly as there are large statistical errors right now) events per month.
On a side note I have yet to see some work which shows that eccentricity can bring out strong field effects.
