From [0, 1/2), the step is 1/2 high.
From [1/2, 3/4), the step is 3/4 high
From [3/4, 7/8), the step is 7/8 high
From [ 1 - 1/(2^k), 1 - 1/(2^(k+1)) ) the step is 1 - 1/(2^(k+1)) high

What is the exact area under the step function?

I'm a little late to come in, and I haven't peeked yet.

The area is a sumation of an infinite sequence:

.5^1*(1-.5^1)+
.5^2*(1-.5^2)+
.5^3*(1-.5^3)+...

Each term is .5^n*(1-.5^n).
Simplify to 2^-n - 2^-2n.

The infinite sum of such a sequence would seem to equal the sum of all 2^-n where n is positive and odd.  So the first term is 1/2 and every next term is a fourth of the previous.  In binary, the decimal would be .101010..., which is equal to 2/3.