Besides the five basic senses (sight, hearing, smell, taste, touch) there is also, for example, the sense of balance. While it may seem that this one is not really about perceiving the external world (as opposed to one's own body), there are cases where it is. E.g., on the inside of a ship with no windows, I can use my sense of balance to detect how strong the waves are. Or I can use it to detect a mild earthquake.
But if we go this route, what if I say that I can use my spring allergies to sense pollen in the air -- does that count as a sense? Taking that further, I could even sense large doses of radiation, since those would make me sick. Maybe there's an argument to be made that in those cases I'm just using other senses to detect something; it is not a separate sense. Then again, in the pollen case, my immune system plays an important role in the detection, and yet we don't typically think of it as part of a sense. Another example is caffeinated vs decaf coffee; I may not be able to taste the difference, but I can detect it based on the effect it has on my brain -- is that a sense?
Going even further: Does part of my brain "sense" mathematical truths?
Is there a clear definition of what constitutes a sense? Does it require a particular type of conscious experience? Or is there a clear definition based purely on biology?