If so, how strong a claim?
To expand the question: if there is a god that can be experienced, does that experience escape science?
And a third question: can humans define something, or work with something that can't be experienced?
What science can explain all experience depends on what you mean by "explain all experience".
Right now, it certainly can't explain every experience--indeed, all scientific literature is full of things that aren't explained, including psychology and cognitive science whose realm includes understanding experience.
But there are no apparent reasons why science couldn't explain everything because, firstly, science is just a formalization of the process of how anyone can come to know anything with confidence, and secondly, our study of the natural world has indicated that while some things are inscrutable, that inscrutability is not at the scale that matters tremendously for human experience (and its neural implementation).
So in principle, science should eventually be able to get it all, should we as a species decide to devote adequate resources to the problem.
If God can be experienced, that isn't enough to thwart the process unless the experience is also extra-physical. If experiencing God happens by virtue of physical changes in our brains, we can (in principle--practice may be very difficult) measure those changes and thus come to a reductionistic account of what it means to "experience God". (In fact, we can already induce feelings of the presence of God through surprisingly crude manipulations.) If there is not only God but also non-material mind, then science (and we) are stuck with descriptions of what people are feeling; we could classify and come to understand these descriptions in great depth, but we couldn't get at mechanism.
Science might also run into some trouble detecting whether it is assuredly God doing things, or whether there are more mundane explanations. If God was annoyed by our inquires, He could hide himself arbitrarily well, or make things arbitrarily ambiguous, or make it blatantly obvious. This isn't because of some defect in science. It's because you actually can't tell the difference between God and no-God if God is careful to intervene in statistically ambiguous ways.
Fortunately, scientists have tons of experience defining and working with things that can't be experienced. Nobody "directly" experiences a restriction enzyme, or a covalent bond, or an isotope ratio, or a Lorentzian manifold. Science requires mastery of abstract concepts whose accuracy and utility are apparent only indirectly. But so do we all--you never directly experience "three", but "three plus two" doesn't flummox most people. A large part of the success of the scientific endeavour can be attributed to our skill (at least after training) at mastering the art of defining and working with things outside direct experience. Adding or subtracting God doesn't really change this (even if you don't experience Him).
The subject is remarkably touchy. Science is a difficult thing to question without ruffling feathers.
The first challenge is to define science. I actually asked a question about this a while back, and the debate was furious. There is a marked lack of agreement as to what science is. In fact, long after that question arrived at Popperian falsification as the key trait (with much disagreement), I was directed by another individual to the worth of Kuhn, who was one of many who argue that "scientific truth is defined by the consensus of the scientific community!"
One key rule for science is that it only claims to explain that which can be modeled. The first question to ask when discussing any interaction between god and science is to ask if the interaction can be modeled. If it cannot, there is no possible way science can claim to explain that experience.
There are interesting corner cases with science because of its marriage to statistics. If you elect to only consider statistically represented analyses as science, any event with a sample size of 1 is simply out of the realm of science because statistics cannot do much with a sample size of 1 (it can do some things, but the inability to reproduce an experiment really puts a damper on science).
As for your third question, we absolutely can define something that cannot be experienced. There are scores of words that have meanings which cannot be experienced. My favorite is the void. You can never experience the void, only "near the void." Mathematics has many others (we have proofs that mathematical concepts exist but cannot name them). However, I encourage you to challenge this claim, because the inner workings of the mind is still not fully understood, and there may be something to these concepts.
Actually, the clearest example of things which cannot be experienced are the Zen Koans, such as the now infamous "sound of one hand clapping." It is clearly defined, but the entire point of the koan is that it cannot be experienced.
Science offers explanatory power; this isn't at all the same as experience; not is everything that offers explanatory power a science.
One can legitimately after all ask how does TS Eliot's The Wasteland attained the significance it has in English literature; and answering this question doesn't involve science.
A Popperian definition of science through the verification principle I find is impoverished; because it describes what it is not; this is rather like trying to prove a negative; one requires a positive description of its aims and methodology; and this to an extent can be described objectively; but this in a sense also falsifies because it doesn't describe the experience itself.
One might describe science in a positive direction; when one describes it at bottom that which we can agree on ie this is a stone, this falls; ie the empirical tradition; but this quickly becomes difficult.
In the Islamic tradition, God is said to be closer than ones jugular vein; but he is hidden behind a veil; In Suhrawardis philosophy/ontology of light, like Ibn Arabi there is an order of revelation; and (I'm speculating here) that understanding and inspiration is part of that order, as Socrates explained or suggested in Ion.
Science as it's now constituted, ie with a Epicurean or positivist philosophy has nothing to do with any kind of divine order; though a hint of an older tradition is sometimes gestured at ie Hawkings 'Mind of God', or Higgs 'God particle'.
Humans work with numbers all the time as a reality; though of course they are not found in nature, as such; in general these are universals; and one might even say universally experienced.
Going with Kant, we synthesise reality; so that reality is a palimpsest of realities.
Badiou, posits four truth procedures - science being one them (the others being art, love and politics); he has a theory of an event; events for him are singular, and unproveable (you know them by their fruits).
I'd suggest that revelation; a revealing is considered as an event, singular by its definition; unrepeatable.
Science cannot, as of yet, effectively explain any experience. Science can explain the interaction of things outside yourself, but it does not include any clear and reliable theory that really explains how anything gets into your mind. We can describe perception, to some degree, but not experience.
And we have a long way to go before we can explain even fairly basic elements of experience such things as 1) how we construct the image of the whole world around us, and of areas of the world we cannot even see, and might never have seen, with real access to only three degrees of clear vision at a time, or 2) how we construct the experience of time passing, even when very little changes around us or even 3) how memory stores certain distinctions between perceptions and not others, while leading us to feel like we remember everything in fairly complete detail.
We do not know whether such theories will come together, or whether the testing mechanisms upon which scientific feedback relies become too unreliable when they involve subjective description.
We already have ways of triggering what most people describe as their experience of God. So if there is a God that can be experienced, science will never be able to tell whether the actual experience is caused by God or not. We already know that it can be caused by other things, that it is not specific enough to identify its proper stimulus. That has nothing to do with God, but with the limitations on the clarity and specificity of our own perceptions.
Mathematics is replete with stuff no one will ever experience, and we determine all kinds of facts about that stuff. So yes, we can define and learn about stuff that is outside of experience. What is the experience of the root of a polynomial? There is none.