If your direct line ancestors increased geometrically, i.e. 2, 4, 8, 16, 32, 64 etc. In 15 generations you should theoretically have 32,768 great (back 15 generations) grandparents. By the time you got back 30 generations you would have 1,073,741,824 grandparents just in that generation. That's well over a billion grandparents. Obviously, how far back you go in 15 or 30 generations depends entirely on the age and the time to when your ancestor was born. If all the mothers had your direct ancestor child at 14 the least time period in which you run up 15 generations would be 15 X 14 or 210 years. But most of us have great-grandparents that lived well over 100 years ago. I have a great-grandparents who were born before the Civil War in the 1850s. A more realistic time period for each generation is around 30 years. So 15 generations would take you back at least 450 years. In my actual pedigrees, if I count back 15 generations, my ancestors were living in the late 1400s.
Now, if you think about this for a minute or two, you will quickly see that identifying over 32,000 people in the 1400s is likely impossible even if you don't take into account pedigree collapse. Pedigree collapse is an absolute reality. It occurs whenever one of your ancestors, knowingly or unknowingly, marries someone to whom he or she is related. The effect of this is that the actual number of ancestors is far less than the theoretical large number. In my line, my parents were second cousins, that is they share a common Great-grandfather. So my pedigree begins collapsing quite early. If you have done your genealogy back a number of generations on all your lines and have yet to identify one instance of pedigree collapse, then you are probably ignoring duplicate individuals in your pedigree.
So when can you expect to reach the maximum number of ancestors? Some estimates put the time frame around 1200 AD. See Wikipedia:Pedigree collapse. If this sounds unlikely, just think of all of the pedigrees tracing their lineage back to the same set of royal ancestors in Europe. In another example, I was talking to one of my friends yesterday about his genealogy and he mentioned that his brother was doing research in Denmark and that it was very difficult to impossible. I was very surprised, because Denmark is one of the most organized and documented countries in Europe. I asked how far back he was looking and he said back in the early 1600s. Oh, well, that explained the problem. It was not a difficulty with the records, it was a difficulty in finding any records before 1600. For example, the earliest records for Denmark in the FamilySearch Historical Record Collections date from 1618.
Given all this, I began to wonder how much effort is going into search pedigree lines that either fall into the category of collapsed pedigrees where the research doesn't realize the relationships or situations where the absolute number of ancestors has been reached and there are no more records? In this regard genealogy is sort-of like collecting stamps, there is is an actual finite limit to the number. You may never have enough resources to collect every single stamp, but you can get the vast majority if you want to spend the time and money. Just like with stamps, genealogy has a practical and finite limit. The further back you go, the more likely you are to be required to spend a lot of time and resources on each incremental acquisition. But there is still a practical and finite limit to how much information you can find on any given line.
OK, so should we all just throw our hands up in the air and quit? No. Very, very few people, given all the researchers in the world, ever reach the practical, much less the theoretical limit of their research activities. But if you don't recognize some of these limits, you might not allocate your resources in reasonable ways. So what is the strategy for extending a line? Especially a line that has reached or nearly reached the theoretical or practical limit. There are a lot of posts and other articles on the so-called brick wall problem. I have a some additional thoughts about that problem, but that is another post.