In his book Scientism and Secularism, philosopher J.P.Moreland lists and discusses five phenomena that naturalistic science cannot account for or explain but which fit comfortably into a theistic worldview. The five are these:
1. The Origin of the Universe: That the universe had a beginning is the consensus view among cosmologists, but if it had a beginning what could have caused it. If the universe encompasses all of space, time and mass-energy then all of this exists only when the universe comes into being, which means that the universe came into being out of nothing. How? The answer to this question lies outside the purview of science.
2. The Origin of the Laws of Physics: As with the universe in general, the fundamental laws of physics exist only insofar as the universe does. Apart from a universe there are no such laws. An explanation of why just these laws exist and have the properties that they do is not an explanation that science is equipped to provide. Science can only tell us what the laws are and what they entail. It can't tell us why they are.
3. The Fine-Tuning of the Cosmos: As we've written on VP numerous times in the past the fundamental forces, parameters and constants which form the fabric of the universe are calibrated to unimaginably precise values such that an infinitesimally tiny deviation in the settings of any one of several dozen examples would make either the existence of the universe impossible or the existence of any kind of significant life impossible. Possible explanations for this extraordinary state of affairs, such as the multiverse hypothesis, even if they're credible, are metaphysical conjectures which lie outside the realm of science.
4. The Origin of Consciousness: Mental states such as holding a belief, understanding a joke, doubting a proposition, feeling pain, sensing red, and recognizing the meaning of a text are phenomena which defy a scientific explanation. On the scientific view there was nothing but atoms, molecules and chemical compounds for eons of time until one day a completely different phenomenon, consciousness, emerged. How does physical matter produce conscious experience? Science has no plausible answer.
5. The Existence of Objective Moral Laws: Science can tell us what is the case in the natural world, but it cannot tell us what ought to be the case. It can explain why people have subjective moral sentiments, perhaps, but it cannot explain how objective moral duties could arise, where they would've come from, why they're binding upon us, and so on. Indeed, any such explanation, even were one possible on naturalism, would be philosophical, not scientific.
These five phenomena come from Moreland's book, but the summaries of them are mine. Moreland's treatment of each is much more detailed than what I've provided here, and he argues that each of these is more compatible with a theistic ontology than any of them are with naturalism.
I enthusiastically recommend his book to anyone interested in the philosophy of science, the explanatory limits of science, and/or the interface of science and theism.