What is the volume of the region in $\mathbb{R}^n$ defined by $0 \le x_i \le 1$ for all $1 \le i \le n$ and $x_i +x_j \ge 1$ for all pairs $1\le i<j \le n$?
For example for $n=2$, this is just the upper triangle of the unit square which is $1/2$.
Alternatively, if $n$ iid uniform random variables are drawn in $[0,1]$, what is the probability that every pair of them add to at least $1$?
I realized Brian Moehring's observation directly gives the answer (without the need for any fancy formulas). If we let $U_{(1)}$ and $U_{(2)}$ be the smallest and second smallest values out of the $n$ uniformly random numbers, then we just want $P(U_{(1)} + U_{(2)} \ge 1)$. But this is a straightforward calculation.
We know $n(1-t)^{n-1}$ is the density function of $U_{(1)}$. Furthermore, given $U_{(1)}$, the rest of the variables are just iid uniform in $[U_{(1)}, 1]$. Thus, given $U_{(1)}$, the probability that $U_{(2)}$ is at least $1-U_{(1)}$ is just the probability that all of the $n-1$ draws land in the interval $[1-U_{(1)}, 1]$, which has relative length $U_{(1)}/(1-U_{(1)})$. However, we have to be slightly careful because once $U_{(1)} \ge 1/2$, the event always holds.
$$Pr(U_{(1)} + U_{(2)} \ge 1) = n\int_0^{1/2} (1-t)^{n-1} \cdot \left( \frac{t}{1-t}\right)^{n-1} \ dt + P(U_{(1)} \ge 1/2).$$
Both of these terms are $2^{-n}$ so the final answer is $2^{-n+1}$.