[t]he likelihood is that it will be solved long before you could have dreamed possible. Think of Auguste Comte who when asked to name any thing that could never be found out instanced the chemical composition of the fixed stars; and almost before his book became known to the world at large, the first steps had been taken in spectral analysis.*Yet there are certainly some questions we won't be able to solve. The problem, of course, is identifying which facts those are.
Traces of the past have been effaced, and so there are some facts about what the past was like that are unrecoverable. In explaining underdetermination to people, I use the colour of dinosaurs as an example. It may just be that the fossil record has not preserved enough for us to figure it out.
And yet researchers claim to figure it out based on microscopic bits responsible for extruding pigment; see the NYTimes article. The Sinosauropteryx, we are told, had "had a head-to-tail feathered mohawk in a subdued palette of chestnut and white stripes."
The story goes on to indicate that other scientists challenge the result, that the data set is small, and so on. And I only ever used the example in a conditional way, to say that the relevant evidence might not exist in the fossil record. I only meant say that this kind of underdetermination will arise in historical sciences. Of course we can't know with certainty which questions will be underdeterminated in this way.
Still, I need a new example.
* I give the full citation and more commentary in an old paper.