Are there any non-finite automata?
All automaton models you'll typically encounter are finitely represented; otherwise there would be uncountably many, which means they are not captured by Turing-complete models. Or, in CS-think, they'd be useless¹.

"Finite automata" are called finite because they only have a finite set of configurations (the input string aside). Pushdown automata, for instance, have a stack that can have arbitrary content -- there are infinitely many possible configurations.



Nota bene: Configurations of PDAs are still finitely represented! In fact, any computational model that falls inside Turing-computability has to have finitely representable configurations, otherwise TMs wouldn't be able to simulate them.




I consciously disregard hypercomputation here for the purpose of this question.