Is it bad to boil water twice in a kettle?
For ordinary tap water, no. The claim that re-boiled water is harmful circulates widely online and does not hold up for the water most people are drinking.
The argument usually runs that boiling concentrates dissolved substances — nitrates, fluoride, arsenic — because some water evaporates. The concentration effect is real but negligible in practice. A kettle re-boiled a second or third time loses a very small percentage of its volume, so any change in concentration is minute, and it starts from levels already regulated as safe in treated drinking water. You would have to boil away most of the kettle repeatedly to produce a meaningful change.
There is a genuine effect on taste, and that is probably what the folklore is really tracking. Boiling drives dissolved gases, particularly oxygen and carbon dioxide, out of the water. Water that has been boiled repeatedly tastes flatter and slightly duller. Tea drinkers often notice it, which is why the standard advice for tea is to use freshly drawn water rather than freshly boiled — the point is the dissolved air, not safety.
The other real effect is limescale. In hard-water areas, repeated boiling and topping up leaves progressively more mineral deposit in the kettle, which affects both taste and the element's efficiency. Emptying and refilling rather than continually topping up keeps the kettle cleaner.
The one situation where it matters is water from a source known to be contaminated, where repeated boiling genuinely does concentrate whatever is present. Boiling sterilises, but it does not remove chemical contaminants — it concentrates them. For that, filtration or a different water source is the answer, not boiling.