How Many Great- (Great...) Grandparents Did I Have?

Ancestry studies based on mitochondrial or Y chromosome assays only detect a single path through our complicated family tree.  For example, my Y chromosome is inherited from my father, and my father's father, who got it from his father's father's (...) father.  A test based on my Y chromosome wouldn't take into account any of the rest of my ancestry contributed from, for example, my father's mother's father's (...) father).

So maternal and paternal lineage data is really about just two branches picked out of our entire family tree.  The results are presented using "haplogroups", which are unique patterns of mtDNA or Y DNA that can be traced back to a single mutation event.  But ancestry is exponential, not linear! Haplogroups only detect a single branch out of a very large number of branches.  And what if my family tree isn't a tree, but a bush?  The scientific term for this is "pedigree collapse".

Assuming a generation time of 20 years, 500 years ago (=25 generations) I could have had 33 million great [...] grandparents! But the population of Europe (I have European ancestry) was only around 60 million in A.D. 1500 (Jan de Vries, compilation of estimates). Assuming no interbreeding of ancestors in the last 500 years, I would have to be related to everyone with European ancestry a little over 25 generations ago. Clearly there must have been some interbreeding, which would imply that I did NOT have 33 million great [...] grandparents.

This probably explains the question better than I can.  

I think the key question is...How many great [...] grandparents did I have? THAT would tell us how informative mtDNA or Y-chromosome haplogroups are.  If I had more grandparents then these tests are less informative.  If I had fewer grandparents they could be quite informative.

I'd be curious just to know ballpark figures. Did I have 10 million or 10,000 great [...] grandparents in A.D. 1500?

I found a study that shed some light on this issue: The Geography of Recent Ancestry across Europe by Peter Ralph and Graham Coop.  According to their results, there was a genetic bottleneck in, for example, the Balkans, around A.D. 1400 and a much larger bottleneck around B.C.  100.  This means that, for example, two average unrelated people in Europe share almost all of the same ancestors as recently as 1,000 years ago!