What is an array that can be iterated circularly an infinite amount of times called? I have a feeling that the answer is either a "circular linked list" or a finite state machine, but the answer is probably a circular linked list.
The answer is circular buffer.
Such structure could be implemented using array or linked list, but you cannot call explicit array-based circular buffer as linked-list.

FSA cannot be "executed" infinitely (type 3 machine only uses fixed input and reads it ones), unless your FSA definition covers Turing Machines.