OSB stands for oriented strand board, which means that three layers of chipped-up wood are laid perpendicular to each other, increasing structural stability. But OSB is not meant to be a finish surface, and that's why this question of painting--a finishing process--is particularly interesting.
Of interest, too, is the fact that OSB is designed for long-term periods of exposure. While OSB manufacturers do not recommend permanent exposure to the elements, OSB was initially seen as a way for remodelers and new home builders to deal with the occasional problem of building delays. OSB stands up well against moisture, so the partially completed home will not immediately disintegrate during building slow-downs.
OSB rated as "Exposure 1" is very good at high moisture over long periods of time.
While not all OSB is the same, The Engineered Wood Association notes that Exposure 1 OSB can be painted. The EWA's main reservation is that some OSB is shipped with a thin wax coating, which would inhibit painting.
The EWA also notes that, due to the prominent visible nature of OSB's strands, it will probably show through on one coat of paint. A heavy primer may be required.
Avoid exposing OSB's edges to moisture; this may allow the OSB to swell and crack.