MO theory methods other than LCAO
Well, this gets into the concept of basis sets.

LCAO implies that the number of molecular orbitals will be the same as the number of atomic orbitals involved.

In fact, we usually use more basis functions than just the simple number of atomic orbitals. We might add diffuse or polarization functions to better describe electron density away from the nucleus.

Moreover, rather than the physically correct Slater-type orbitals:

$$R_l(r) = A(l,\alpha) r^l e^{-\alpha r}$$

We usually use Gaussian-type orbitals to express the radial component.

$$R_l(r) = B(l,\alpha) r^l e^{-\alpha r^2}$$

Now GTOs don't express the radial part of the wave function well -- they decay too quickly as you get far from the nucleus. So we use a linear combination of multiple Gaussian orbitals to approximate the physically-correct Slater terms. Still, it's much faster to use a set of multiple GTOs.

Beyond STOs and GTOs, people do use other types of orbitals. For example:


Wavelets
Plane waves
Muffin-tin