With an extremely liberal definition of phases of matter...
The atoms in an amorphous solid are aligned in a rigid disorderly structure, instead of a regular lattice like an ordinary ("crystalline") solid. Liquid crystals are intermediate between solids and liquids; the atoms are held in place, but are free to rotate.
In many materials, there are actually a variety of solid phases, each corresponding to a unique crystal structure. These varying crystal phases of the same substance are called "allotropes" if intramolecular bonding changes or "polymorphs" if only intermolecular bonding changes. For instance, there are at least nine different polymorphs of ice that manifest under different temperature and pressure conditions. Diamond, graphite, and buckministerfullerenes are allotropes of carbon.
When quantum fluids are cooled down enough, they gain superfluidity, and can even phase into supersolids. Quantum effects manifest themselves to produce extremely strange things.
A quantum fluid is a cluster of electrons moving together as a whole, sort of like BEC or fermionic condensate, but under extremely high pressures and low temperatures they may condense into a superfluidic state.
http://upload.wikimedia.org/wikipedia/en/4/46/Supersolid_phase.png
Superfluids are... well a fluid taken to the extreme. Superfluids have NO viscosity. Superfluids have NO entropy. Superfluids flow with NO friction. Superfluids have infinite thermal conductivity. Superfluids can conduct electrical current with the absence of Voltage.
Supersolids are superfluids in the solid form, I don't know much about those. I heard they can be pushed through walls (?).
Under extremely high pressure, ordinary matter undergoes a transition to a series of exotic phases collectively known as degenerate matter. These phases are of great interest to astrophysics, because these high-pressure conditions are believed to take place inside stars that have used up their nuclear fusion "fuel", such as white dwarfs and neutron stars.
Degenerate matter is has a high enough density that the dominant contribution to its pressure arises from the Pauli exclusion principle.
Theoretical quark-gluon plasma is same idea as plasma but with more heat and pressure, and constituted of quarks and gluons.
Metallic hydrogen consists of a lattice of atomic nuclei, protons. It occurs when liquid hydrogen is compressed far enough. Metallic hydrogen is present in tremendous amounts in the gravitationally compressed interiors of Jupiter, Saturn, and some of the newly discovered extrasolar planets.
Neutronium is a colloquial, not-well understood, and often misused term describing the highly compressed phase found in the core of neutron stars.
White dwarf matter is the same idea, but in a white dwarf core.
Strange matter is the next step after neutronium. It is theorized that after neutronium inside a neutron star is put under sufficient gravity pressure, the individual neutrons break down and their quarks form a "strange matter."
Of course the later ones are totally theoretical, only solid, liquid, gas, plasma, liquid crystal, amorphous solid, BEC, fermionic condensate, quantum fluids, superfluids, supersolids, metallic hydrogen, and possibly quark-gluon plasma have been created in the lab.
http://physicsweb.org/articles/news/8/9/15/1
This recent article is very interesting, a substance created in the lab that freezes as it heats.
