After boiling water in an electric kettle, should I empty it?
Your boss is somewhat right. Probably not wrong enough to argue with him.

There are several types of scaling that can be caused by mineral content in a water supply. One of the most common is called "temporary hardness" or "carbonate hardness." At high or neutral pH, calcium and magnesium carbonates have quite low solubility in water: 139 ppm for magnesium carbonate, and a mere 6.7 ppm for calcium carbonate, both at 25°C. (NOTE: Until starting this reply, I hadn't realised how little agreement there is on the solubility of calcium carbonate. Every source that I checked agrees that it is quite low; but no two sources agreed to even one sig. fig.!)

However ordinary, clean rain water often has a pH as low as 5.6, through absorption of atmospheric carbon dioxide. (Actual acid rain is even lower!) When such water runs through deposits of these very common minerals, some of them dissolve through the reaction:

$\ce{CaCO_3 + H_2O +CO_2 &lt;-&gt; Ca^{2+} + 2HCO_3^{-} }$

and the similar one for magnesium. Calcium bicarbonate has an aqueous solubility of 166 g/L at 20°C -- some 13,000 times more soluble than the carbonate. Of course, the dissolution process from hard rock is slow, and most runoff will not actually be saturated.

The problem is that our reaction, above, is an equilibrium; anything which tends to drive it back to the left will cause precipitation of the insoluble carbonate. One of those things is just atmospheric outgassing; this is how stalactites and stalagmites form. But a much faster method is boiling the water. That decomposes the bicarbonate ion, and rapidly drives off the freed carbon dioxide gas. The resulting precipitate of calcium carbonate is called "temporary hardness" because, as you have observed, it is easy to remove with any mild acid. Some other forms of scaling are much less forgiving.

Now, if your water hardness problem is severe, a lot of carbonate will start to precipitate as soon as the kettle is heating up. This is why another commentator suggested that it is too late to do anything after the kettle has boiled: that the scale deposits have already grown. However, there are two complications. Firstly, the supersaturated calcium carbonate solution doesn't necessarily all precipitate at once; the scale crystals may continue to grow for some time. I can't say just how long, but a process whereby large, hard crystals tend to cannibalise small, soft ones can continue for weeks.

Secondly, you will not see hardness that bad from mnay common water sources. Well water, maybe, but most municipal supplies will already be softened to the greatest degree that it is economical. If the water hardness problem is mild, some of the calcium carbonate will precipitate immediately, but some (more than 6.7 ppm!) will stay dissolved in the very hot water, until the solubility falls with falling temperature and more of it precipitates.

Long story short, your boss's concerns are not unreasonable. On the other hand, as you note, temporary hardness is easily cleaned off, and only temporary hardness is related to boiling. On the third hand, it's his kettle and he's the boss, so do as you're told!