Several others pointed out problems with the scenario you've presented. Some have thought your scenario sounded a bit like radical skepticism, <PERSON> demon. Presumably, those folks were imagining that the rest of the universe looks as if the laws of physics continue to apply, but, per the scenario in the OP, perhaps they do not. This sounds a bit like the <PERSON> brain scenario; if you're not familiar, check out <PERSON> recent "Why Boltzmann Brains Are Bad" or his pop-level book From Eternity to Here. Others have pointed out that we can make measurements of objects beyond the solar system and that, to the best of our measurements, the laws of physics continue to hold beyond the solar system. Fair enough.
I'm going to take a different tact. It's long been noticed that the galactic rotation curves -- i.e. the velocities of objects as a function of their distance from the center of galaxies -- goes towards a constant for large distances. That's not expected on standard accounts of gravity, given all of the mass we can observe. 484
The standard response is to say that there is more mass than we can observe -- this is one way physicists infer the existence of dark matter. But notice that we could propose a different hypothesis. Perhaps, instead of invoking dark matter, at very large distances, we need to modify the standard equations for gravity. This leads to a view called Modified Newtonian Dynamics (or MOND) in which, at large distances, gravitational physics changes.
What might be of particular interest to you are the so-called "Pioneer Anomalies". There are two objects that have effectively left the solar system -- Pioneer 10 and 11 -- and both exhibited unexpected motions after passing approximately 20 AU (or 2,000,000,000 miles). For some time, physicists thought the Pioneer Anomalies could be due to new physics, as with MOND or other speculative theories. Later, the anomalies were explained by an anisotropic radiation pressure caused by the spacecraft's heat loss -- in other words, not by new physics. But this provides a real world example of what you might have in mind -- new physics might be detected when we leave our solar system, perhaps physics our present day observations only hint at (as with the galactic rotation curves). What this would require is that the effects are consistent with all of our present astronomical observations, but significant enough so that, once one leaves the Sun's gravitational well, one would begin to see something surprising! 921