every time the same water is boiled, the concentration of dioxins goes up, enough so to be a health concern,
boiling water twice makes water poisonous
Health effects of boiling of water
Boiling some water kills off any biological organisms in it that might cause you harm (Reference, Reference Reference). Once something is dead there is no point trying to kill it a second time, so boiling once just before use is sufficient.
Boiling water will also reduce the concentration of volatile contaminants - making the water safer (Reference)
If you boil water for a very long time, you may reduce the total volume of water to say half, this would double the concentration of any dissolved compounds that are less volatile than water (Example for reference purposes). Normal use of electric kettles doesn't reduce the volume of water sufficiently to make this a concern.
Dioxins have low volatility and low water solubility (Reference). So they only appear in sediment, if you filter out the sediment you should remove most of the dioxins. If you boil your water for a long time and condense the steam and drink that (i.e. distilled water) you would reduce the quantity of dioxins ingested.
1: Dioxins (if any) in your water are probably more likely to be present in sediment in the bottom of the kettle than dissolved in the water. Modern kettles turn themselves off before significant volume of water is boiled off and so don't produce high concentrations from low concentrations. Rinse your kettle once a week if worried.
2: It isn't the ordinal number of boilings that is important, it is the reduction in volume of the water - which is related to boiling time. This isn't a significant factor for normal use of domestic kettles.
So I conclude the claim is wrong.