It has been reasonably argued that the Universe as we know it is a state machine. It has also been argued that the Universe is infinite. If the Universe is a state machine and the Universe is infinite than the answer to the question has to be "no".
In other words, the question is philosophical rather than computational.
Infinite State Machines
A Turing machine is such a beast. But it is also hypothetical.
IndeterminateNumberOfStates State Machines
Before I die I must use that as a class name.