I usually throw away the remaining water in the electric kettle before refilling it if the water is more than about a day old. In theory wouldn't boiling kill any impurities the old water might have? How long would the water stay drinkable in the kettle (assuming you boil it again before drinking)?
Maybe I'm wrong, but there is no safety risk at all when boiling old water. There is however something like taste. The reason the old water is discarded is because, after boiling, it lacks oxygen and will taste stale. This is also the reason why coffee is made with 95ºC hot water, or why Moroccan tea is poured from above (to oxygenate)...
So, as long as you boil it before drinking it should be safe.
There shouldn't be much microbial growth in clean water (as there is, after all, nothing for the microbes to eat). Eventually, you'll get some from e.g., dust falling in, but that'll take a while. And, as you say, boiling will kill them; that is after all one way to render untreated water is rendered safe.
The one thing that boiling won't remove—actually, it'll concentrate—is any impurities leaching from the kettle. How much of a risk this is depends on what the kettle is made of (and, I suppose, how acidic & corrosive your water is). Glass (non-crystal) and stainless are probably of no concern. Aluminum and copper might be. Some plastics are.
There are only three possible things that could affect the 'safety' of water -- contamination, removal of disinfectants, and removal of other compounds.
Now, if you're bringing it up to boiling each time, and you haven't put in water from a tainted source, and you're only putting in water (and not using it to heat other liquids, like soup), contamination isn't a big problem.
The other issue is the disinfectants used. It could be a chemical (iodine, bromine, chlorine, chloramine) but some places, mostly in Europe use ozone (ie, an alternate bonding of oxygen).
It's possible that some of them might come out over time ... unfortunately, when I took environmental engineering, I don't remember if they specified how it comes out, only that there's an assumed half-life. (chlorine is so common because it stays useful as a disinfectant for longer). If you keep refilling the pot, and refilling when it gets low without pouring out the stuff in the bottom, it might come out of solution and collect in the bottom of the pot. There are some concerns about stuff that can bond with chlorine, but you'd have to have those other things in the water to start with, and it's not going to be any worse than sitting in the pipes for long periods.
Ozone, however, might be a problem. Because it's a disolved gas, it'll come out when the water is boiled, so your water won't have a sterilization agent in it. If you then let it sit, it could possibly grow something ... but if you didn't have the pot contaminated in the first place, it's likely not a problem. And if you're bringing it to a full boil at least once a day, it's not going to have that long to develop problems.
But you're right in that longer times sitting without boiling is a problem ... I'd say the problem time is likely more than a day and under ideal situations could go for months, but I don't know how long, and that could depend on the other issues in the area. (eg, you find that loaf of bread that managed to go off, but you've opened it releasing spores into the air ... and then you go to work ... I'd change the water, even if it had been fresh and boiled that morning). I'd be even more concerned in areas that use ozone for sanitation, as it'll come out in a boil, and even if you're not bringing the water to a full boil, has a relatively short half-life as a sterilization additive.
As for the removal of other compounds ... just as BaffledCook suggested, yes, other stuff could come out of solution ... but the thing is, you'd have drank it anyway. As it's a fixed amount in over time, there'd be no net ingestion change for you, just that it might be slightly different compounds (eg, iron ions will react with oxygen to precipitate out). But assuming it did concentrate, you can just solve that problem by not using the last cup or so from the kettle. If you ever get down that far, dump it out, rinse it out, and refill.