First off I'm not going to get into an esoteric philosophical debate about what a human sense actually is. Whether it only relates to making sense of external input from the environment around us or making sense of internal body functions, suffice to say we need both to function as living beings.
So how do they stack up? Lets start with the traditional five.
Sight or Visual perception is the ability to interpret the surrounding environment by processing information that is contained in visible light. Seems pretty straightforward, you can either see it or you can't. So that's one discrete sense? Well you have two different types of sensors in your eye, Cones and Rods. Rods are designed to work in low light and measure light intensity. Cones are designed to work in bright light and detect colour. There's three distinct types of cones, one for each primary colour. So that's 1-5 sense(s) depending on how you count them.
Hearing is the ability to perceive sound by detecting vibrations for which we have a pretty nifty set of sensors in our ears. So that's 1 sense.
However our ears do much more than that. They provide you with your sense of Equilibrioception or balance enabling you to orientate yourself to the gravitational field, so you can tell up from down. So that's 1 more sense.
Touch is a complex sensory system made up of a number of different receptors. At least five different types of nerve endings relating to touch have been identified covering, heat, cold, pain, itch, pressure. So that's anywhere between another 1-5 sense(s).
Taste, in humans taste buds are concentrated in the tongue and top of the epiglottis. In 2010, researchers also found bitter taste receptors in lung tissue, believe to be an evolutionarily adaption to fight chest infections. Five types of taste bud have been identified to date sweet, bitter, sour, salty and umami (pleasant savoury taste). So that's another 1 -5 sense(s).
Smell, is still a poorly understood area with recent research suggesting humans can distinguish anything between a thousand trillion and 80 million scents. In many vertebrates smell is subdivided into the main olfactory system for volatile chemicals and the accessory olfactory system for fluid-phase chemicals. In water-dwelling organisms smell and taste is often combined into a single chemical sense and its often said that in humans taste and smell are combined to give us a sense of flavour. So for arguments sake lets call smell 1 sense.
Proprioception is the ability to tell where your body parts are, relative to other body parts. When you close your eyes and touch your nose your using this sense. Unless your drunk, in which case it can be impaired. That's 1 more sense to add to the list.
Stretch sensors tell you when your bladder and bowel is full, which is pretty essential when you think about it. So lets clock up another 1.
Finally thirst monitors your hydration levels and tells you when you need to drink. Likewise hunger tells you when you need to eat. So there's another 2 senses. Although they sometimes get combined into a single generic sense.
So unless my maths is out that gives you at least 9 and perhaps as many as 22 senses.
Its also been suggested that in addition to all this Magnetoception, the ability to sense magnetic fields and the ability to perceive time should also be considered as human senses, but were straying into the realms of pseudoscience here so lets stick with 22 tops for now. Which probably explains why we have so much trouble making sense of the world around us at times.