A classic question in knot theory is: Given a smooth knot in the 3-sphere, what surfaces in the 4-ball can it bound? In symplectic geometry, a natural question is: Given a Legendrian knot, what Lagrangian surfaces can it bound? Whereas any smooth knot can be filled by an infinite number of topologically distinct surfaces, there are classical and non-classical obstructions to the existence of Lagrangian fillings of Legendrian knots. In particular, a polynomial associated to the Legendrian through the technique of generating functions can show that there is no compatible embedded Lagrangian filling. Legendrians that admit generating functions will always admit compatible immersed Lagrangian fillings, and I will describe how this polynomial also gives information about the minimal number and indices of double points in compatible immersed Lagrangian fillings. In addition, I will describe some constructions to realize minimally immersed fillings. This is joint work with Samantha Pezzimenti.