  We study the connection between the affine degenerate Grassmannians in type
$A$, quiver Grassmannians for one vertex loop quivers and affine Schubert
varieties. We give an explicit description of the degenerate affine
Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of
type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex
loop quiver provide finite-dimensional approximations of the degenerate affine
Grassmannian. Finally, we give an explicit description of the degenerate affine
Grassmannian of type $A_1^{(1)}$, propose a conjectural description in the
symplectic case and discuss the generalization to the case of the affine
degenerate flag varieties.
