I'd like to challenge your question.
What is a sense?
Are you looking for a complex organ that integrates with the central nervous system? Or just any part of your body that can be used to determine the property of something in the outside world? For example, you can use your toenails to test whether a hot plate is of sufficient temperature to melt dead keratinous biomaterial, like your nails. This would count as an integrated sense for a specific physical chemistry.
Or do you think a sense is a thing that detects a singular property of the world? For example, do you discriminate proprioception from gravity sensing, even though it may be performed by the same organ? Historically, each classic sense is limited to one property, but this I think is outdated. For example, skin is not only capable of performing touch and texture detecting ("tactile" sense), but also heat, pain, and other things like sensing noxious poisons such as bee stings.
Also, do senses have to detect stimuli in the external world? Plenty of sensing goes on internally, both in interstitial spaces and fluids in your body, on the surface of cells, as well as within cells, and the variety of cellular and sub-cellular equipment is hugely various. Should light receptive cells only count as 'vision' or 'light receptive' when they are found in the eye, or can light-sensitive cells found deep within the body can also be classified as part of the visual sense? There are many animals and tissues in biology where light-sensitive receptors are not exclusively found in eyes.
These are only 3 things you did not consider. There are many other questions you must answer along these lines to have a clear, at least temporarily operational definition that can be used to make distinctions to answer what things can count as a separate, countable sense.
Just as a proof of concept, I can list more than five senses off the top of my head to show you that the "five senses" concept is woefully outdated in my view, and I'm sure some others would agree. I will pick some weird examples just to demonstrate the complications that the above questions illustrate.
(considered a sense for chemistry? or a collection of senses?)
- Olfaction (smell, as carried out by neurons in the nasal epithelium; e.g. smell of vanilla, and smell of bad food)
- Gustation (taste, as carried out by neurons on the tongue; e.g. salt, sugar)
- Antigen chemosensing (chemical sensing, as carried out by, for instance, immune antigen receptors on B cells)
- Hormonal signaling chemosensing (chemical sensing of hormones such as insulin, as carried out for instance by myocytes)
- Starch sensing? (amylase in saliva can be used as a test for digestable starch)
Visual system, at the retina?
- Visible light (sensing electromagnetic radiation on the order of a few hundred nanometers in wavelength)
- Internal methanol sensing (the visual system as a sensor for methanol, which disproportionately affects myelin surrounding the optic nerve)
- Pressure sensing (see phosphenes)
The vestibular system
- Gravity sensing
- Motion sensor
- Head position sensor
- Spatial orientation
- thermosensation (touching a hot kettle!)
- Nociception (pain sensing)
- allergen sensing
- sensor for gamma rays, X-rays and UV light (indicated by radiation burns, development of skin cancer, sunburns, etc.)
Bones and muscles?
- Kinesthetic and bodily proprioception
- mental pain
- mental or spiritual distress
- sense of self and other, including friendship, power, place in social hierarchy, reputation, companionship
- motivation and love (oxytocin, dopamine, etc. in limbic systems and other neural correlates)
I'm sure some would agree, and some would disagree about the specific cases I provide. Thus the definition of senses, or sensing, seems to be opinion-based or at the very least very sensitive to an agreed-upon operational definition, for which there is none.