Is there an automaton with the given properties?

Question

Let us assume that automaton A has the property that at most one transition enters each state. I wonder if such an automaton can recognize a language containing arbitrarily long words?

Answer 1

The answer is yes. Consider the automaton $1 \xrightarrow{a} 1$, where the alphabet is $\{a\}$ and the unique state $1$ is both initial and final. This automaton recognizes the language $a^*$ which contains exactly one word of every length.