At the end of the 19-th century physicist thought they had figured it all out. All of elctromagnatism in just 6 (Maxwell) equations. What a victory. Newton's laws of motion and gravitation describing how things move including all of the celestral bodies. But what a surprise when relativity and in particular quantum mechanics messed up that neat clean picture. Today there are many who believe all of physics can eventually be reduced to a few fundamental laws of physics. But even this extreme form of reductionism does not imply we can stop the scientific endeveaour there and then.
Perhaps the most canonical example of what some people called non-science is an attempt to describe the physics inside a black hole. Almost by definition, the black hole is an object from which nothing ever escapes, so it seems utterly futile to try and understand what goes on inside it. If only because one's theories can not be based on observations. That was before the modern insights that Hawking radion can eventually radiate out the information stored in a black hole. I guess the debate isn't settled yet, and the picture is somewhat more complicated (for observers that remain outside the event horizon nothing really ever falls into the black hole, instead it freezes in time on the horizon and so the information is more like stored on the horizon rather than inside of it.)
But there are other questions one can ask pertaining the possibility to even finish physics (and therefore science as physicist like to believe.) Can we ever find explanations for everything? Can we find explanations for why certain physical constants have the value they have? And if so, can we explain why the laws of physics are what they are? There always seem to remain questions unanswered. This infinite regression reminds of a child's game: "Dad, why is the sun warm?" "Well, because it consists of a big fire", "but why is the sun on fire dad?", "well because there is nuclear fusion that makes it really hot", "but dad, why is the there ...."
A different problem asserts itself when we try to explain how the first reproducing molecules came into existance. Any such theory will rely on a random, lucky first event that created such a molecule. Yes, all ingredient must be present but we still need to reason like: "there are X possible events per year, times Y years times the probability P of that event gives us Q=XYP% chance life came into existance in the 3 billion years available to the earth. How big must Q be before we are satisfied, is 0.0001% ok, or do we require more like 50%?
And then there is chaos, which presents perhaps the biggest blow to the cherished notion of predictability in science. It asserts that certain systems are unpredictable by any practical methods even if wee know the laws of physics exactly and know the initial considtionals to a *very* high degree of accuracy. It's a fascinating topic and it shows that even knowing the laws of physics at their most fundamental level may not be very helpful in understanding the phenomena that result from it. Instead, we ought to be looking at the problem on a larger, less "fundamental" scale. Ineresting structure can still emerge by changing the magnifying glass. The primary example is thermodynamics and its relation to statistical mechanics. The collective behavior of large quantities of chaotically moving particles can result in predictive phenomena described by quantities like pressure, temperature etc.
When can we say that we truly understand something? Take the brain. When can we declare victory? Even if we can replicate a brain in all its detail in a computer (including consciousness etc), do we understand the brain? How many levels and how much detial of explanation do we need?
And then there is this annoying mathematical theorem: Within any framework build on a finite set of axioms one can formulate questions that remain unanswerable within that framework (Godel.) I am not sure how relevant this turns out to be, but it ought to make us a bit humble.
So, were will science end up in the end? Never quite dead I hope. Perhaps, at a fundamental level we will end up with an infinite number of theories which are all consistent with the observations we have (it seems string theory is heading that way right now.) But I predict we will always be busy with improving theories at a practical level. Improving our ability to predict the weather, or earthquakes for instance. Up till now the mathematics we have used to describe physical phenomena has been surprisingly elegant (Riemannian geometry for general relativity, Hilbert space theory for quantum mechanics, etc.) and moreover it was almost always invented by mathematicians before its application to physics (modern string theory being the exception.) This to me seems highly suspicious. I see no good reason why the mtahematics describing physics ought to be elegant. It raises the suspicion there may still be a whole lot of "ugly" physics out there which has not been filtered through our "elegance sieve", the high hanging fruit so to speak. I predict there is much, much more of that than we can even start to imagine. No, science has not come to an end, it has just started.