The maximal sum of interior angles is achieved by drawing a very small triangle somewhere on the sphere and then declaring the inside to be the outside and vice versa. The sum of the interior
and exterior angles is necessarily always $3\times 360^\circ$ and since one of these sets cannot sum to less than $180^\circ$, the opposite one cannot be
more than $5\times 180^\circ$. —
Henning Makholm Oct 24 '11 at 23:52