The linear optical properties of semiconducting carbon nanotubes are dominated by quasi-one-dimensional excitons formed by single electron-hole pairs. Hence, the nonlinear response at high pump levels most likely leads to the formation of exciton complexes involving several electron-hole pairs. Such complexes would therefore play an important role in, e.g., lasing applications. We demonstrate here that the biexciton complex is surprisingly stable for nanotubes in a wide diameter range. Theoretical predictions for the signature of such states in pump-probe spectroscopy are presented.