There are couple of factors at work when re-steeping green tea: temperature, time, and the quality of the tea (the size & way it's been processed). Green tea is supposed to be brewed with water that's been brought to boil and allowed to cool to 167 - 176 degrees Fahrenheit (though many people simply heat the water to that point or what they eyeball as hot but not boiling). Green tea is typically steeped for 1 - 2 minutes. And green tea, especially loose leaf, but also with higher quality tea bags, has furled and sometimes rolled leaves. The lower temperature, the short steeping time, and the curled leaves mean that flavor is still left within the leaves even after multiple steepings. The leaves continue to expand and release in subsequent baths. I don't, unfortunately, know any of the exact science--what's being released when, etc.
However, a possible alternative to boiling & brewing individual teas when done or simply using more tea (the typical choice when brewing larger quantities), you can bring your water to temperature, steep, pour it into a preheated or heat retaining container, and repeat the cycle until you've brewed your multiple cups in one go. As far as I know the tea leaves don't need to 'rest' in between brewings.
The reason green tea can be re-steeped is also the reason most black teas can't. Black tea is subject to boiling water for a longer period of time (typically Western use is 3 - 5 minutes; some cultures favor an even longer steep for the deeply bitter tannins). The hotter and longer steep pulls more flavor out of the leaves faster. Some black teas, especially high quality whole-leaf, can be re-steeped, however. Most bagged black teas--and a lot of loose leaf--can't.
The heat & steep time is dictated by the processing of the tea once it's picked. Green tea has minimal processing--pluck, wilt, shape & dry. Black tea is allowed complete oxidation (called fermentation). Fermentation breaks down chlorophyll, releases tannins, and forms many of the taste & aroma compounds that typify black tea. Oolongs and other varieties are subjected to varying degrees of fermentation. The size & shaping of the tea leaf--allowed to remain whole, broken, rolled, or (for most bagged tea) cut into fannings and dust, also affects how tea should be initially steeped and whether it can be re-steeped. Highly rolled oolongs, for example, often hold up to more steepings than green tea and are even said to taste best only on the 3rd or further steeping.
Best Answer
You could try making tea with loose leaf tea and a strainer (rather than tea bags). I have an inexpensive teapot with an insert. It is big enough to make 2-3 cups. There is still a little of the tiny tea flecks in the bottom, but if you pour carefully they stay in the pot and you can have a no-residue cup of tea.