Question

Does ammonia have filled `pi` orbitals if it doesn't have any `pi` bonds?

Answer 1

Well, look up the MO diagram of `"NH"_3`. I get:

So yes, it has filled `pi` orbitals. Any `pi` orbital is doubly-degenerate, meaning that there are two identical energies. The HOMO-1 here are your two `pi` bonding molecular orbitals.

Note that they aren't really making `pi` bonds.

The HOMO-2 and HOMO-1 orbitals are what were involved in making the three `sp^3` bonds with the surrounding `"H"` atoms. The HOMO holds the nonbonding (slightly bonding) pair of electrons.