Does water "go bad" in this sense?

It doesn't go bad, but it does change the taste.

When water is just sitting there, water evaporates, but most things dissolved in it don't. Then, each time you boil it, the steam causes additional water to escape leaving the same amount of dissolved stuff in there. So, the concentration of dissolved stuff keeps going up.

Dissolved oxygen also decreases when you heat or boil water, changing the taste.

Finally, even if your kettle is stainless steel, it can eventually rust.

If you don't do it too often this might be a good idea for another reason: when you pour out the water (and preferably give your teapot a rinse) you also pour out concentrated chalk residue and any other particles that didn't boil.

Chalk is the white stuff that forms around in your boiling pot after a while. It isn't bad for you at all, but it is bad for your boiling pot. By emptying it completely every once in a while you give your appliance a longer life (independent from how long the water has been in the pot). Chalk doesn't disappear when boiling. So, if you boil more and more water without ever getting rid of the little bit of water at the bottom, it has nowhere to go and the concentration of the chalk will rise and it will stick to the pot after a while. If you empty out the water, the new water will have a lower concentration for a while and you get less residue.

How bad the chalk is depends on the quality of the water where you live. Also, if it really has been a long time since you last used it, I would also clean it because dust still comes in. It's probably not dangerous to drink, but can affect the taste and isn't nice.

As far as I know it does not go bad, but there is a good reason not to fill your kettle completely when you fill it from the tap.
Partly filled kettles come to the boil in less time. Also costing less in energy, which can be a concern for some.

So fill the kettle part way, say to the amount of water you actually use and you will spend less time waiting but no more time filling.

I often refill my kettle after using it till empty, having it ready to run and already at room temperature when I want to have an other hot drink.

Here I do not notice much difference in taste between fresh and re-boiled water but that may differ between locations.

Your water goes bad because of microbiological growth.

Your tap water is not pure water. There is a lot of stuff in it. And your environment is not sterile and simply cooking it to 100°C for a short amount of time does not sterilize it. You have to cook it for 10 minutes and your container needs to be airtight after.

Potable water does not "go off" ever*

I wouldn't even consider actually changing the water in a kettle unless I was going to be away from the house for more than a week, for which I'd simply empty it before I left & fill it again when I got back.
Any bacteria acquired through the spout is going to be negligible, and you are flushing the system through with fresh water every time you boil it [or once a day, if you insist on filling it unnecessarily to the top each day, which does nothing but waste energy]. Even if you leave some in the bottom each time, it's not the 'same bit'.

Hard water - high calcium carbonate concentrations leading to limescale & furred-up kettles - will taint the flavour anyway, whether you start with fresh each time or not. The difference in flavour between 'new' & 'mixed with the old' is going to be negligible. Allowing limescale to build up doesn't really hurt anything except your electricity bill & the kettle element which will eventually burn out from overheating.

The solution to both flavour & electricity bill is a water filter.

I live in a very hard water area but grew up in a very soft water area. I consider a filter an essential item, not a luxury.

*assuming it is not later contaminated. This is supported by building regulations, which allow you to cap a pipe without ever needing to allow for the water remaining in that unused section ever being drained or flushed.

Ok, so to put it in simpler terms. Lets say you have a bucket of marbles,(Water) and sand (Dissolved solids). Each time you boil the water, two marbles are taken away until eventually there is no water, but dissolved solids instead and this can really mess up the taste, and mess up your pot/kettle in general. So the best thing to do is trust your instincts on this one and just dump out the old water.

My recommendation is to boil it about 4 times and then ditch the batch and refill the kettle.

I work in drinking water quality chemistry at a municipal water authority. The "Potable" water at your tap is treated with chlorine. For the water to be safe from microbial growth, there must be a sufficient "residual" of chlorine. However, the chlorine breaks down over time, for example, water that has sat stagnant in a main for about a week or more will lose its Cl residual, and must be flushed to waste because it is not "potable" anymore.

Heating water breaks down and evaporates the chlorine almost completely. That kettle of warm, non-chlorinated water sitting on your counter is a prime breeding ground for microbes, some of which could be disease-causing. Always dump out the kettle when you're done with it, never drink (or cook with) water from your home’s hot-water heater, and re-boiling water will NOT necessarily make it safe again.