how long will yeast live once pitched?

hehehe "rack off."

If you kept removing the old wort and adding new wort, then you would probably have some living yeast in there pretty much indefinitely or until the fermentor was full of old, dead yeast, and there was no more room.

If you just added sugar without removing any liquid, you would reach the upper end of the yeast's alcohol tolerance and it would go dormant/die/autolyse.

 

There would still be viable yeast in the fermentor. How much or how ideal the situation is I couldn't say.

 

In theory, the yeast will "live" (reproduce and die off) forever, given the right conditions. If you are adding wort, you have to remove some, or all, of the fermented beer. The yeast can only tolerate so much alcohol.
Just be sure to oxygenate your wort before you add.
I've heard that the Jim Beam distillery has been using the same yeast strain since its founding in the 1800s. Its just been continuously fermenting in a vault.
That being said, I'm not sure how long you would want to keep the same yeast strain active for brewing beer. Some of the qualities will likely change over time. Most notably will be the flocculation.
Note: I don't have much experience with this. I'm just regurgitating what I've heard. Good luck!

 

This should work out nicely I would imagine. At some point the 1056 might not be doing as much as you'd like, but if you are adding dregs and such it shouldn't hurt it. I have a Lambic I made last September with yeast I cultured from blackberries that took my 1.054 OG to 1.006 in 3 weeks. I added dregs to it periodically as well. Come February the beer was starting to oxidize due to being in a 6 gallon carboy and only 5 gals of beer (and I always want to sample). I brewed up a Saison and used an extra .5 gallon to top off the carboy. Over the course of 6 weeks the then 1.005 Lambic took the added 1.042 Saison and the balance of the Lambic down to 1.002. I'd say that the yeast are still viable after 6 months (YMMV). The key as some have stated would be the alcohol. If you are doing this with a 1.042 beer first, a 1.055 next, and maybe a 1.060 for number three, and only a gallon at a time, the alcohol shouldn't be too high. You would end up with a type of solera sour after it is all finished. Give it a shot. Even if the 1056 craps out on you, you'll have a fairly large starter for house sours with all those dregs in there, and given time in the barrel, the bacteria and wild yeast will work through it.