I like tea, and I like the British way of making tea, as described by such experts as George Orwell, Douglas Adams, and my spouse. An important part of this process is steeping the tea in a pre-warmed mug or pot. When I can, I fill a mug with boiling water, wait for it to heat up, pour off the water, promptly fill it again with a fresh pour of boiling water, and only then add the teabag.
Using two pours of hot water rather than one improves the flavor of the resulting tea immensely. If two is better than one, than perhaps three might be better than two. Should I preheat the mug twice? This is a question that can be answered with science.
To a very rough approximation, a ceramic mug and the water in it weigh about the same (about 300 grams). The weight of the teabag (about 3 grams) is small enough to neglect. The specific heat of water (4,184 J/g˚C) is about four times the specific heat of the in the mug (~900 J/g˚C). Thus, they will reach equilibrium at about four fifths of way from the initial temperature of the mug to the initial temperature of the water. If the mug starts at room temperature (20˚C) and the water is boiling (100˚C), this means the tea will brew at about 84˚C.
Pre-heating the mug once with a second pour of water means that the mug starts at 84˚C before the brewing water and tea are added. The water still starts at 100˚C and the specific heats are unchanged, so now the equilibrium is about 97˚C. That’s a big difference!
Preheating the mug a second time with a third pour of water repeats the process again. Now the mug starts at 97˚C, and the third pour brings it up to about 99˚C. That’s a small difference!
I conclude that on theoretical grounds, preheating your mug by using a second pour of water raises the brewing temperature by about 13˚C, enough to result in a substantial improvement in flavor. Preheating the mug again by using a third pour of water raises the brewing temperature by only about another 2˚C, which is not enough to make a noticeable difference unless you have a more refined palate than mine. Preheat once if you can, but you can stop there.
This result has been confirmed experimentally.