  We introduce \emph{k-positive representations}, a large class of
$\{1,\ldots,k\}$--Anosov surface group representations into PGL(E) that share
many features with Hitchin representations, and we study their degenerations:
unless they are Hitchin, they can be deformed to non-discrete representations,
but any limit is at least (k-3)-positive and irreducible limits are
(k-1)-positive. A major ingredient, of independent interest, is a general limit
theorem for positively ratioed representations.
