In his book The language wars: A history of proper English, Henry Hitchings states that there are eight countries which do not have an official language: Bosnia-Herzegovina, Costa Rica, Eritrea, Ethiopia, Pakistan, Somalia, the UK and the USA. Throughout the book he provides extensive references, but not for this one. My spider-sense is tingling: Australia does not have an official language.
The most convenient and authoritative source I can think of is the CIA’s World Factbook. It lists Bosnian, Croatian and Serbian as official in Bosnia-Herzegovina, Spanish in Costa Rica, Tigrinya, Arabic and English in Eritrea, Amharic in Ethiopia, Urdu and English (‘lingua franca of Pakistani elite and most government ministries’) in Pakistan, and Somali and Arabic (‘official, according to the Transitional Federal Charter’) in Somalia.
For the UK, it lists ‘the following … recognized regional languages’: Scots, Scottish Gaelic, Welsh, Irish, Cornish.
For the USA, it states that ‘the US has no official national language, but English has acquired official status in 31 of the 50 states; Hawaiian is an official language in the state of Hawaii’.
And for Australia, it simply lists languages spoken: English 76.8%, Mandarin 1.6%, Italian, Arabic, Greek, Cantonese, Vietnamese (each with 1.1-1.4% of the population).
To be fair on Hitchings, he does preface his list with ‘At the time of writing’, and his book was published in 2011. Those six countries might have declared their official languages since then (the World Factbook was last updated last month), but he should have known about Australia.