To take just one example, consider the idea of the multiverse. This is the hypothesis that physical reality is like a vast foam in which every bubble is a discreet universe. This is a completely untestable hypothesis for which not only is there no empirical evidence, there couldn't be, by the nature of the case, any empirical evidence. In other words, the multiverse hypothesis is a metaphysical, not a scientific, hypothesis.
Another problem that seems impervious to empirical evaluation is the problem of consciousness. Tallis writes:
Beyond these ... problems there is the failure of physics to accommodate conscious beings. The attempt to fit consciousness into the material world, usually by identifying it with activity in the brain, has failed dismally, if only because there is no way of accounting for the fact that certain nerve impulses are supposed to be conscious (of themselves or of the world) while the overwhelming majority (physically essentially the same) are not. In short, physics does not allow for the strange fact that matter reveals itself to material objects (such as physicists).There is, as far as I know, no physical theory that explains how a physical process such as might occur in the brain can generate something like the conscious experience of redness. Nor is there any physical description of what, exactly, the experience of redness actually is. Nor is there any physical explanation for how meaning, what philosophers call intentionality, can be generated by chemicals moving across neuronal membranes in the brain. How does the movement of atoms produce something like a meaning? How can one describe a meaning, for instance the meaning of this sentence, in terms of chemical reactions in the brain? Science has no answer to these questions.
Read Tallis' relatively brief essay for more on how physics is shot through with, and quite dependent upon, philosophy.