However weak the challenger's "power" is, a sufficiently long series of trials would get you the one in a million, if the "power" really existed. Indeed, if the "psychic" could say in advance how much more accurate he or she was than chance, then it would be possible to arrange a test such that there was only a one in a million chance of failure --- if the "psychic" really did perform at the level claimed. That would be nice. Unfortunately, all claimants are people who have not tried to find out to what degree their abilities differ from chance. This is, I suppose, because people who do try to find this out find that the answer is 0, and so they don't become claimants.
Nonetheless, I make this suggestion to richard for what it's worth: if you can get them to state in advance how accurate they are, then it's fairly easy to do the maths to see how many tests they have to do so that they have, not just a one in a million chance of succeeding by luck alone, but also such that they have only a one in a million chance of failing --- if they really are as accurate as they claim.
Whether that would satisy davidsmith73, I cannot foretell: for someone who can complain about the alpha value of an experiment per se is clearly not mentally equipped to get a firm grip on a statistical argument.