I am interested in running cellular Automata to run a simulation on city growth. Which GIS software can run this out of the box, or via plugins?
I was able to do this in ArcGIS using Repast's Agent Analyst Extension a couple of years ago. It allows you to do agent-based geosimulation, CA included. It uses Java or a subset of Python called Repast Python. Not sure if it's still supported, though. It goes by Repast Simphony now. Here is a walkthrough on creating cellular automata models to get you started.
Another tool for ArcGIS is the cellular automata tool that comes with Hawth's Analysis tools. I don't have any experience with that one though.
You may have a look at the GAMA platform:
GAMA is a simulation platform, which aims at providing field experts, modelers, and computer scientists with a complete modeling and simulation development environment for building spatially explicit agent-based simulations (...). GAMA provides the modeler with the ability to use arbitrarily complex GIS data as environments for the agents.
For CA, see for example this tutorial.
I don't think any COTS product will have a built-in tool for CA.
You can write your own logic or use a combination of built-in tools to achieve the result.