What can we learn about physics by doing experiments at the end of the heliosphere? My question is very naive, as I do not know cosmology and astrophysics.  I am asking it just for general curiosity. All of us know that Voyager I is now going out of heliosphere, and now sending data regarding this very boundary. Can we expect some interesting / new physical discoveries?  By this I mean, certain results which can give us deeper insights on physical laws or open new avenues of work. 
(I believe, I am not very clear here and shall get a few angry comments for spamming. Advanced apologies for that. I remember the 1998 observations on distance–redshift relation for Type Ia supernovae. I want to know whether we can expect some data which are similarly striking.)
 A: I'm not an expert on the Voyager missions, but as a theoretical particle physicist/cosmologist I can tell you that nobody really expects the Voyager probes to discover anything "fundamental" in these fields. We've already got a rough idea of what the interstellar medium (ISM) is like from numerous astronomical observations (I'm sure an astronomer could tell you more), and basically the energy scales are too small to probe the sort of new physics that, for example, the LHC explores. On the other hand the energy scales are too large to probe the sort of frontier that cosmology explores. So the ISM is well in the range of physics we think we understand.
There is a small chance that the Voyager probes could discover something like MOND, if careful tracking of their orbits does not follow our understanding of gravity. For a long time (certain) people thought this might be the case for the Pioneer anomaly, but very careful studies of the solar heating of the spacecraft showed that the anomaly was mundane - only a misunderstood systematic error, not new physics. So it is conceivable but very unlikely, along both theoretical and experimental lines, that the Voyager probes could discover modified gravity. If it does happen then that will be tremendously exciting!
Hopefully an astronomer will come on here and tell you what they do expect to learn from Voyager. Presumably something about the conditions of the ISM.
A: The conclusion to a recent paper by Webber and McDonald (requires access, preprint available on arXiv) on the V1 data says,

If the galactic cosmic ray (GCR) and anomalous cosmic ray (ACR) intensities continue to remain at their present levels, then indeed this "heliocliff" region displays many of the properties of a "classical" heliopause, perhaps a much more impressive barrier to inward and outward transport of energetic particles than would have been anticipated
There are many interesting effects that could be observed beyond this barrier as the heliospheric neutral Hydrogen is mediated in the local interstellar medium (LISM). These include possible low level modulation of GCR and effects on the plasma of the LISM. Future observations at V1 will hopefully settle many of these issues as the spacecraft proceeds further into this uncharted region.

That pretty much says it all: we have found the edge of our solar system (the heliopause) and we hope to see what the galactic cosmic rays (primarily hydrogen ions) do to the plasma outside the solar system.
A: We can learn about interstellar magnetic fields and why they cause the heliosphere to have a heliotail that is like a four leaf clover.
eg.
http://www.space.com/21913-solar-system-tail-first-photos-unveiled.html
ps. Not a voyager result though.
