What do you mean by "testable"? There's a pretty big spectrum of "untestable" ranging from requiring too many trillions of dollars to requiring more resources than this planet has to requiring equipment or energy levels we don't know how to produce even in theory to requiring stuff we're pretty sure is impossible like looking beyond the range of the observable universe or requiring more energy than is contained in the observable universe.
And at what point in its development and study do you abandon a theory that nobody has yet invented a way to test?
Except the question remains unanswered. "Untestable" is basically a grey area. There's "untestable right this minute, but in 5, 10 or 20 years it will be testable", then there's "we can probably test it in 100 or so years..." then there's the "it would take 1,000 years or more (maybe never) to test it".
Theories like String Theory tend to fall into the first, possibly dabbling with the second option. Researchers are already starting to work on ways to test string theory, with promising results. It's quite likely that within 20 years, we might be able to do just that. While things like the many worlds interpretation of quantum mechanics or multiverse theories tend to fall on the "maybe in 1,000 years, but probably never" category.
If you mean we're going to be hearing about the Many Mystical Worlds of Quantum Physics from Deepak Chopra types for the next 1,000 years, I'd say "not very" is a reasonable answer.
An even better question - is this the most interesting of all possible untestable theories?
String theory's critics are sure that it isn't. I've even heard it said that it only has the prominence it has because a noisy generation of physicists elbowed out competing projects.
If you don't have a testable theory in the short term, should you carry on trying to refine that theory in the absence of evidence, or should you spend more time on competing theories that perhaps have more chance of being open to experimental verification in a reasonable time frame?
The list of open, unsolved, and unexplained issues in physics isn't small. Continuing to prioritise one theory and gambling that it works out eventually is a classic sunk-cost error.
A large chunk of physics is theoretical in nature. Sometimes you simply cannot experimentally validate a phenomenon, so you instead propose a theory that attempts to explain it using physics. Then in the future, when the phenomenon is testable, your peers determine whether your theory was correct or not.
Most times an experiment isn't possible/practical to perform, but the experiment could at least be described. The Higgs particle "If we build an accelerator that accelerates particles to energies pf X electron volts, we should see traces of it" must have sounded almost like a thought experiment when the Higgs boson was theorized. Same thing for e.g. the General Theory of Relativity and the Mercury passage that was one of the first validations of it.
Clearly if you propose a theory that requires an experiment so advanced (or circumstances so rare) we can't hope to do it in 20 years, it's still a valid scientific theory. What about 50-100 years? What if it requires technology so advanced it's nearly unthinkable that we will ever attain it? This is when it becomes a philiosophical gray area. It's not a clear cut case what is verifiable and what isn't, since the theory is valid before our ability to verify it.
You could also derive the Atomic nature of matter through math though, as infinitely devisable sets end up with paradoxes that don't match up with reality (think banarch-tarski)
You can do all this without an idea how to test it.
Given the knowledge available 2000+ years ago, atomism was a surprisingly good idea. Yah, they got a lot of the details wrong, but that basic idea that reality is made up of basic "building blocks" that combine into the larger structures we can observe is basically correct.
The philosophers that invented atomism probably thought about how the might see their atoms directly. I wonder how many "crazy" ideas were dreamed up that sounded impossible, that are now easy experiments to do today.
Theory is important, even if it sounds impossible today, because we have a history of redefining what is "possible" throughout the history of science.
Well, there is the case of Aristarchus who came up with the heliocentric model of the solar system about 1800 years or so before Copernicus. His model was rejected because of experimental evidence -- the theory predicted stellar parallax which observation at the time could not detect! (Of course, we can detect it now that telescopes have been invented -- the stars are just so much farther away than anyone could have possibly believed at the time.)
Cosmology has been an uncontroversial part of physics for quite some time now, and most of its major claims can never be tested in the same that other physical theories can be tested, barring some "creation of a universe in a lab" scenario.
Then why ask philosophers? If you don't know what the other side of the moon looks like because rockets have not been invented yet, how could a philosopher possibly help?
Maybe not. It's not completely clear that, for example, string theory could ever be experimentally verified, since it operates completely at a level below all the manifestations of our observable universe. (AFAIK, I'm no physicist)
What technology can you imagine that could ever prove or disprove a hypothesis of our universe being only one facet of a multiverse? The apparatus would necessarily exist within the closed system.
But the multiverse theory is in competition with other theories, some of which make the same predictions. If two or five or seventeen theories all make the same predictions, do you pick one? Which one? If you want to pick the simplest theory, what criteria do you use to figure out that one theory is simpler than another theory?
And at what point in its development and study do you abandon a theory that nobody has yet invented a way to test?