Is a Petri net without places well-defined?
A net without place does have "state", or a marking in proper terminilogy; as that is a mapping from the set of places into $\Bbb N$, technically this does exist even when the set of places is empty (as a kind of empty relation).

A net with one place and one transition can even change state. If the transition has an outgoing arrow to the place, the transition can add tokens into the place. (Tokens can be created, they aren't physically moved from one place to another.) Effectively the behaviour will be that of a upward counter.