In most (all I have seen) work on cellular automata, the state of a cell is from a finite set. Such as a cell can take value from a binary set, or from among few colors. Can we define the state to be something more complicated such as a function of the coordinates of the neighbor cells? (assuming there is a initial coordinate value give to at least one cell) So for example in 2-D if my neighbor to the left has coordinate $(x,y)$ my state will take a value $f(x,y)$ where $f$ is same for all cells.
Although Wikipedia mentions a finite set of states (http://en.wikipedia.org/wiki/Cellular_automaton), I'm sure you could find some interesting models that use infinitely many states. As you suggested, each cell could contain an integer, a continuous value between 0 & 1 (shade of grey), etc.
The idea of basing cell contents upon cell coordinates is interesting and unfamiliar. Of course, you'd have to also include some other inputs in the transition function if you want the pattern to change from step to step. This would create patterns that vary based on their location, which sounds fun to explore.
I'm hardly the expert on these matters (my only point of contact being an artificial life class a while back) but a bit of googling found an interesting application of infinite-state cellular automata: "lattice-gas" cellular automata which apparently are used to solve PDEs by discretizing the diffusion equation. I'm strongly suspect there are more such applications, but like I said, I don't know much about them.
Note another book which says somethig rather obvious namely that the classic cellular automata lie at one extreme end of discretization with space, time and state being all discrete in CAs, while classic PDEs are at the other end with all three continuous. It also notes that there's no reason not to consider mixtures of these; it also gives additional synonyms you can google for continuous-state (but discrete time and space) cellular automata: coupled map lattices and lattice dynamical systems; on a quick look both terms seem to produce a fairly rich literature. So whether these are (properly) called cellular automata... it depends who you ask, I suppose. (Also I could not help myself LOL at the comment on next page in that book [which is at its 3rd ed.] "Obviously, this field is tempting for computer freaks".)