Are there materials with more than two states that can be used in circuit design? Are there atoms or materials that have 3 or more states that can be used in circuit design? Can you provide any examples of such materials, and their possible applications to circuit design? And is there a maximum number of states that such a material can have?
I am interested in systems where each part has three possible states, so that e.g. a combination of two such systems have $3^2 = 9$ possible combined states:
0 0
0 1
0 2
1 0
1 1
1 2
2 0
2 1
2 2

 A: Take any material and three of its energy levels. 
Three-level systems also go by the name of qutrit. The linked article also links to some research results. 
The number of states is completely irrelevant for theoretical purposes, it's just like calculating with decimal or binary calculations. As long as you can manipulate and entangle systems across many qubits, qutrits or whatever, you can define a universal gate set. It would maybe look differently, but unless any system is built, this is not really interesting.
There are however some physical differences (see the linked article), which might make them an interesting object to study.
Edit: Maybe, I'm misinterpreting the question, so let me add some references of quantum computation with qutrits:
http://arxiv.org/pdf/quant-ph/0506031.pdf This proposes a design of a qutrit quantum computer with ion-traps. 
http://www.mis.mpg.de/preprints/2014/preprint2014_9.pdf This is a preprint of a scientific reports article investigating the geometry of a qutrit quantum computer (proving essentially that the optimal circuit is the shortest path on the $SU(3^n)$)
Another interesting approach might be 
http://arxiv.org/abs/0804.0272 
where they use qubits, qutrits, etc. to reduce the number of gates - but this seems not to be in the vein of your question, as it ultimately operates not only with qubits or qutrits, but rather with qubits and qutrits. 
A: A typical particle that can have three states is the hydrogen atom in the ground state (or in general in an ℓ = 0 state), if J = ħ i.e. if the total spin (of the nucleus and of the electron) is ħ. This state has three possible projections of the spin on whatever direction, +ħ, 0, and -ħ.
Another typical particle with spin ħ is the deuterium nucleus, s.t. it has also three values of the spin projection on whatever direction.
Well, you can make your choice !
Good luck!  
