Zero.
I hadn't realized that until I read it recently. And the reason is so simple it should have been obvious to me, but I always thought about it as progressive 50/50 blends each generation.
Why? You have 23 pairs of chromosomes. Except for the Y chromosome, for each of those you get a copy of one of the two your mother carried, and likewise from your father. Ignoring crossover for a moment, that means your mom had 23 chromosomes from her mother, 23 from her father, and you'll get about 11 or 12 from each of them. And about 6 from each of your great grandparents. And 3 from your great-greats...you see where this is going.
Crossover happens at a rate of about once per chromosome as a rough average, so you're really dealing with 92-ish building blocks rather than 46 chromosomes, but that just pushes it back by one generation.
By the time you're at 10 generations and 1024 ancestors, you only have 92-ish blocks from that group of 1024. You don't have a single base pair from the overwhelming majority of them. Go back 13 or 14 generations, and you have nothing from 99% of them.
The Y chromosome, and your mitochondrial DNA, follow a different set of rules. They're 100% from the all-male or all-female branch of your tree.
This strikes me as 100% logically solid and 100% counterintuitive. I like that.