Tricky Uniform Distribution Problem

[T50_Omaha8]

I'm trying to use game theory to evaluate the optimal number of shills for an auction house to employ.

I'm having much more trouble with a seemingly simply lemma than I thought I would:

Given n points randomly chosen along the interval [0,1] (openness or closedness irrelevant), what is the expected value of the largest point?

I can't come up with anything prettier than saying it is a sequence described by:
c(n) = a(n)/(2^((2^n)-1))
where a(n) is defined recursively by
a(1) = 1 and
a(n) = (a(n-1))^2+(2^((2^n)-2))

c(1)=1/2
c(2)=5/8
c(3)=89/128
c(4)=24305/32468
...

Beat, brag, or variance?

Or am I missing something simple?

Note that I'd prefer to keep this as a run-of-the-mill equation I can do calculations with, not a recursively defined sequence.