Numerical evidence suggested $\pi(x)$ is always less than $\mathrm{li}(x)$.
Littlewood proved that $\pi(x) - \mathrm{li}(x)$ changes sign infinitely often, but the smallest $x$ s.t. $\pi(x) > \mathrm{li}(x)$ is currently not known. The smallest such $x$ is Skewes' number There is a crossing near $e^{727.95133}$. It is not known whether it is the smallest.
The problem possibly might be solved by some clever method other than naiively computing $\pi(x)$, but I don't see why this argument doesn't apply to the other answers.
