Cofty: I am interested in examples of ways to know things by means that are not available to the scientific method. I strongly suspect such examples exist but I can't think of any.
I am an atheist, but I do not believe the scientific method is our only means to knowledge or even the most reliable means. The scientific method is ultimately empirical - it relies on the testimony of the human senses. The problem is that the human sense organs were not designed to give us universal truth, they evolved to give us very limited information about our environment - just enough to avoid falling off a precipice or getting eaten by a sabor-toothed tiger! For example: the actual EM spectrum is enormous - encompassing everything from radio waves to gamma rays - but the portion of it that we can actually perceive as visual light and heat is almost infinitesimal. Furthermore, human senses are subject to all sorts of illusions and hallucinations. They just aren't that reliable beyond mundane day-to-day purposes.
Science is strongly aligned with the philosophy of logical positivism. Logical Positivism is a philosophical position originated by Ludwig Wittgenstein and his Vienna circle of philosophers. Briefly, LP states that that which cannot be perceived by the human senses or mechanical extensions of those senses (such as microscopes) is, for all practical purposes, nonexistent; in otherwords: absence of evidence is evidence of absence. This is frankly an absurd proposition! For thousands of years humans possessed no ability to perceive bacteria, but those bacteria nonetheless made their existence felt in the form of terrible diseases! I can understand Wittgenstein's motives: he was fighting religion and superstition; but his commitment to empiricism blinded him to its inherent limits.
So if the senses are unreliable, what can we depend upon? The answer to this question was discovered nearly 2500 years ago by the philosopher Socrates, whose thoughts were recorded in the books of Plato. Socrates realized that since the human senses were limited, unreliable and easily deceived, we must rely on something else that each of us has but few humans ever develop: logic. Logic can allow us to discover things about the universe that our sense organ never could. A good example of this is from the realm of mathematics. It is impossible for any sense organ to perceive something like negative or imaginary numbers, and yet simple reasoning from the axioms of mathematics tells us that these entities exist beyond any doubt. But logic is not limited to mathematical and metaphysical truth. Albert Einstein arrived at most of his conclusions on the nature of space, time and light not through the scientific method - at the time most of his theories would have been impossible to test - but purely through mathematical deduction and thought experiments that he carried out in his imagination! Einstein's theories have since been empirically verified.
When it comes to universal truths - Truth with a capital 'T' rather than the mundane facts of every day existence - logic has proven itself time and again to be the superior method. Truth arrived at empirically is always contingent, always vulnerable to the discovery of new facts which might contradict it (and in fact science is always changing its mind on things). Truth arrived at through logic can never be refuted so long as the logic is sound. Notice that, unlike science, mathematical theorems are never refuted. The body of mathematical discoveries has always grown, but nothing is ever discarded as outdated. Calculus did not replace trigonometry, it grew out of trig as a logical extension of it.
I don't mean to demean the scientific method as useless. Clearly it given us many fruitful discoveries. But as a rationalist, empiricism will always take the backseat when I'm trying to figure something out. This post is long enough already so I won't go into details, but I will say that the exercise of reason has allowed me to solve many longstanding metaphysical question - such as the nature of consciousness and its relationship to matter - that I don't believe scientists will ever be able to arrive at using their methods.