As far as the Pirahã have related to researchers, their culture is concerned solely with matters that fall within direct personal experience, and thus there is no history beyond living memory.
The language is claimed to have no relative clauses or grammatical recursion, but this is not clear. Should the language truly feature a lack of recursion, then it would be a counterexample to the theory proposed by Chomsky, Hauser and Fitch (2002) that recursion is a crucial and uniquely human language property.
The people do not count and the language does not have words for precise numbers. Despite efforts to teach them, some researchers, such as Prof. Peter Gordon of Columbia University, claim that they are incapable of learning numeracy. His colleague, Prof. Daniel L. Everett, on the other hand, argues that the Pirahã are cognitively capable of counting; they simply choose not to do so.
UPDATE: As Wikipedia says, they are in serious danger of extinction with only 200 left. To help them, perhaps donate to The Nature Conservancy, which works to preserve habitat in Brazil?