How long does it actually take to extract tea compounds from leaves using the boiling method? So, I'm looking at the intersection of multiple brewing style of tea extraction, and the boil-it-to-death style of brewing, and I would like to know about how long does boiling actually work to extract more flavor from already-steeped tea leaves, and when is it just extra time.

In general, I usually steep my tea once or twice, with the time and water according to the temperature on the directions.  It makes pretty good tea, especially since I usually use a lot of leaves, and brew lightly the first time.  In any case, once the tea is spent - I save the leaves up and make a third and sometimes fourth cup by boiling the tea leaves to death.  

The tea isn't bitter, since so many of the tea compounds are already extracted in the first few brewings, but it isn't weak because the leaves are brewed for extra heat and time, and also there's a larger amount of leaves (the used leaves from several brewed cups making one boiled cup), both combine to make a drinkable tea.

So my question is, does anyone know how long does this boiling needs to take?  Any suggestions for when the boiling has extracted as much tea as is coming out in that round (due to saturation of the water or equilibrium in the compounds between the water and tea), and when the boiling is unnecessarily long? 

It's clear to me that there's a point after which more boiling isn't helping - some teas will take two or three boils to exhaust what's left in the tea, and longer boiling for the first brew doesn't necessarily mean the third less strong, so clearly there's some equilibrium being reached with that amount of that specific tea and that amount of water.  So maybe there's no need to boil it for an hour, if all the flavor that will come out is done in fifteen minutes - or else maybe there's no need to keep it simmering for three or four hours, when 45min to an hour is what it actually takes before the tea strength stopped changing.

If anyone has an idea of when boiling the tea stops extracting flavor, I can cut down on the extra time I leave it simmering because I'm not sure.  I would really appreciate any insight others have on the topic.
I agree with the above comments that it really depends on the tea. The best way to find out is to boil the leaves let's say for 5 minutes in not too much water, then strain out the leaves. Now boil the leaves again for 5 minutes in fresh water. Does this second boil result in coloured tea liquor? Then this confirms that boiling it for more than 5 minutes is worth it. If so, then go for the next 5 minutes boil it for the next 5 minutes and so on, until you can confirm that you can't get any color and flavour out of the leaves.

Let me know the result!