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What is the largest "allowed" seed for a PRNG to not give any extra power to a deterministic machine?

I think there must be some confusion in the problem setting. Given an input of $n$ bits chosen uniformly at random, there is no algorithm to compress it to something whose length is on average ...
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How many random bits does this algorithm use on average?

I don't have a formal proof, but I expect the expected value is $\lg n + O(1)$ bits. In particular, heuristically I am expecting something like $\lg n + 2.9$ bits or so (but I don't trust that the ...
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