If you take left over yeast from a wash and dump it in new mash, and repeat, eventually the yeast stop reproducing due to extensive budding scars. So after 4-12 batches people buy new yeast... if not right away. But I was just thinking, why is that so different from sour dough? Bakers can keep thier yeast alive indefinitely by feeding them flour when they are hungry. And if you think about it logically yeast have to be able to reproduce indefinitely, otherwise there wouldn't be any yeast. They would have gone extinct in weeks if they could only handle several batches.. I think they only excessively reproduce by budding under the harsh circumstances brewing beer. But I wasn't able to find that specific information online, nor what it is about brewing beer specifically that kills them. I'm wondering if they can mitigate budding damage by reproducing normally. And I wonder if to get more out of a line of yeast one could do some simple tricks like instead of tossing them straight from one mash to the next, put them into dough in between?
Yes you can use yeast indefinitely.
However doing so takes some work. Keeping a strain unmutated requires a lab.
A specific generation is usually only good for 2-3 fermentations. This is the equivalent of pitching wort right on top of the previous batches cake. There's little to no growth if not oxygenated and the yeast go right back to feeding. But like most living things they die in time and the cycle triggers seem to be the peaks and valleys of available oxygen and food.
You can get several more runs by washing slurry of dead yeast and pitching a growth amount which is the same by volume as pitching from new yeast.
The problem with keeping a culture and growing it over and over is mutations. In even a few generations you will influence the yeast by your growth methods. Yeast will begin to favor the sugars you give it. It may become more or less flocculat based on how you wash. It may start producing esters not typical of the strain when denied nutrients or otherwise stressed.
This is why yeast makers cryo store the samples and grow each run from a single loop sample of the original strain.