In short, no.
The five traditionally recognised senses are hearing, smell, sight, taste, and touch. This is thanks to Aristotle, a Greek philosopher well-known for getting science mostly wrong; for one thing, he thought that the Sun went around the Earth. To his credit, his five senses are the ones most easily characterised, but these can be subdivided, and there are other senses that don’t fit within these categories. And before you ask, no, telepathy does not count as a sense.
A “sense” is defined as a physiological ability to detect a stimulus; usually, this comes down to what can be detected by a single type of receptor. Pain, for example, is measured by nociceptors. It doesn’t fall under the umbrella of the Aristotelian senses, not even touch, because pain receptors are different structures altogether to pressure receptors. Temperature detection is also a separate sense, with different receptors for hot and cold temperatures.
"There are many non-traditional senses. Your sense of balance is one, as is sensing of where your body parts are (unless you’re missing some of them)."
There are many non-traditional senses. Your sense of balance is one, as is sensing of where your body parts are (unless you’re missing some of them). Hunger, thirst, and lack thereof are all senses. Stretch receptors inform you of if you have breathed in too much air, or if you’re bloating from intestinal gas.
A sense of time passing is also a sense. In a sense. Neurones across your brain work together to measure the passage of time, but these take a back seat when something more exciting is happening. That is why “time flies when you’re having fun”. Another sense that doesn’t use a single receptor is recognition, where you might subconsciously recognise an item or situation that you may or may not have encountered before. This is partly responsible for déjà vu, as well as the “sixth sense” of something being off.
Even within the traditional senses there are many subdivisions of detectable stimuli. For example, try to remember the last time you ate spinach: If you thought it tasted horrible, you probably have an active TASR38 gene, which allows you to taste phenylthiocarbamide (which is found in spinach). The TAS2R proteins are responsible for detecting bitter chemicals – like phenylthiocarbamide, but also theobromine from chocolate or quinine from tonic water. TAS1R proteins deal with other tastes, mainly salty, sweet, sour, and umami; the recently described “savoury” taste that is responsible for the flavour of meat, or tomatoes.
Incidentally, taste buds hold a record in the human genome. Of all the protein groups, taste buds have the most points of genetic variance overall. This is why you may hate the taste of liver and onions while your friend might happily scoff them, as well as why some people have a “sweet tooth” while others will skip dessert. The group with the second greatest variance is the MHC molecules, which are crucial in preventing disease. A minor effect is that smell receptors can detect these, and will subconsciously inform you on if you are related to that person. No, really, accidental incest can be prevented by having a good smell. Luke Skywalker and Princess Leia probably had olfactory problems.
Many animals have senses that most humans do not. Pigeons can detect magnetic fields, while bats and dolphins can navigate using echolocation. Fish have gas bladders which help them figure out their current depth, and some are capable of detecting nearby electric fields.