Interestingly, it would appear you can only guess at most a million (1048576, to be precise) different things like that!

Think of it this way. At every stage, you look at all the knowledge you have, combined with the previous questions and answers you've received, and generate some question. Then you get one of exactly two answers. So you can reach at most 220 different results, no more.

Even if you allow yourself to use some (presumably weighted) randomness to generate your questions, the expected number of correct guesses is (at most) 1,000,000.

But I don't think anybody really believes this line of reasoning. What's wrong, then? Perhaps we adjust ourselves according to what we know of the person who chose the object, or maybe we "read" them for more information. Odd.