It is the Takbīr, but Insha'Allah or Alhamdulillah could also be used.
In many threads on reddit that show videos of current conflicts in the Islamic world people mock Muslims for saying Allahu Ackbar on any occasion. If someone asks why it is said, the answer is usually 'that's what they say, just like 'Oh Jesus''.
Is there a historic reason why Allahu Akbar is used so often? Especially compared to other religious denominations and their phrases?
As an Arab and former muslim, I can explain the meaning of the phrase now. Basically it's a reminder that God is the Greatest being in the universe. In times of happiness, it's more celebratory, like hallelujah, when you're in a state of awe, Allahu Akbar is also said, but differently, with a more solemn tone. When you're sad, you say it to remind yourself that God is out there looking out for you. It's essentially an Arabic Muslim catch-all. The purpose, religiously, is to always remember who is the most powerful being in the world, Allah.