The other day I got in an argument over what constitutes a "scientific question," after I invoked thought experiments in an attempt to illuminate some features of a model. Is a question that cannot be tested even in principle a scientific question? And thus, if it isn't, should scientists think about it at all?
Forced to take a point of view, I want to offer the possibly only intelligent words the former German chancellor Helmut Kohl ever uttered "Entscheidend ist, was hinten rauskommt." - "What matters is what comes out in the end."
I don't care very much about somebody's personal philosophy or what -ism they classify themselves under, I care whether it's an approach promising to lead to progress. Discarding a question as unscientific when it concerns extreme, and potentially untestable, limits extends the requirement on a theory to be scientific to the method how to get to a good theory. And with that cuts off frequently used methods, not only thought experiments, but also demanding mathematical consistency in regimes we have not tested. I simply think limiting allowable questions to those practically testable constrains our imagination and thus our opportunity to find answers.
One can certainly take such though experiments and the questions they raise for more important than they are. Steve Giddings for example claims that our failure to solve the black hole information loss paradox is akin to the failure of classical electrodynamics to explain the stability of atoms (see this talk or this paper). Well, the only difference is that Bohr's very existence was evidence for the stability of atoms, whereas we have yet to see a single black hole evaporate. Nevertheless, the information loss problem indicates clearly a lack in our understanding of Nature, and, wanting to understand, physicists have turned it inside-out and upside-down for decades. However, whether that eventually leads to something besides loads of papers yet has to be decided.
A more fruitful though experiment was of course Einstein's chase after a photon. And then one should not neglect that thought experiments are not only inexpensive but also fun.