Humans can use language to refer to and describe endless varieties of situations, thoughts, ideas, and topics, including hypothetical situations and events that never happened. This capacity, referred to here as topic-openendedness, is a key feature distinguishing human languages from animal communication systems and any theory of language and language evolution should account for it. Recursion, the mechanism that provides language with the capacity for discrete infinity, can account for the fact that languages can create an infinite number of sentences from a finite set of words and rules. But it cannot account for the openendedness of the contents of those sentences. Therefore, the importance attributed to recursion as the sole mechanism that is uniquely human is overrated. We suggested that a key factor in explaining topic-openendedness is the nature of the linguistic symbols, the words, specifically their ability to extend their meanings beyond their basic meaning, to other, novel semantic domains, by means of cognitive processes such as metonymy and metaphor. Meaning extensions allow language users to apply a finite lexicon to an infinite number of situations and topics, and play a crucial role in explaining topic-openendedness.