How Many Fair Bits Fit in Eight Outcomes

Let $\Omega = {0,1}^3$ carry the uniform probability measure, so each of the 8 outcomes has probability 1/8. Every random variable here is a function defined on that $\Omega$.

What is the largest $n$ for which you can define $X_1, \dots, X_n$ on $\Omega$ that are iid Bernoulli(1/2)? Report 1 if two independent such variables are impossible, and 0 if even a single Bernoulli(1/2) is impossible.

Answer format: enter an integer.