  We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur
functions called the almost symmetric Schur functions. These functions
simultaneously generalize the finite variable key polynomials and the infinite
variable Schur functions. They form a homogeneous basis for the space of almost
symmetric functions and are defined using a family of recurrences involving the
isobaric divided difference operators and limits of Weyl symmetrization
operators. The $s_{(\mu|\lambda)}$ are the $q=t=0$ specialization of the stable
limit non-symmetric Macdonald functions $\widetilde{E}_{(\mu|\lambda)}$ defined
by the author in previous work. We find a combinatorial formula for these
functions simultaneously generalizing well known formulas for the Schur
functions and the key polynomials. Further, we prove positivity results for the
coefficients of the almost symmetric Schur functions expanded into the monomial
basis and into the monomial-Schur basis of the space of almost symmetric
functions. The latter positivity result follows after realizing the almost
symmetric Schur functions $s_{(\mu|\lambda)}$ as limits of characters of
representations of parabolic subgroups in type $GL.$
