The answer is Papua New Guinea, which has around 840 living languages.
This is truly surprising in a country of barely 8.25 million, but the reason is the country's formidable topography. Its mountainous terrain meant that even tribal groups living in close proximity to one another might have little or no social or economic interaction and therefore they developed languages which are completely incomprehensible to one another.
The one language that unites the whole country is pidgin, "Tok Pisin", a no-frills lingua franca. "How are you?" is "Yu stap gut?", "I don't know" is "Mi no save". A beard is "mausgras", an aircraft is a "balus", and if it crashes, the locals will say "Balus, i bugarup".
Australia, too, has many indigenous languages. Since the country has been settled for a very long time, with tribal groups scattered across a vast landscape and isolated from their neighbours, it's reckoned that about 250 languages were spoken when Europeans arrived, with around 800 dialects.
About 90 per cent of those languages are regarded as endangered through lack of use, or loss of elders able to pass them on.