The greatest ancient civilization was the Romans.
The Athenians at Salamis probably had the greatest influence on history. Both Western and Eastern history would have been completely altered.
Culture, military power and dominance? That is a difficult one to answer but I'd go for Sparta. It had a military power and political influence far in excess of any comparably sized state and that was entirely due to its militarized culture.
Athenian democracy is an incredibly interesting topic. I would have liked to have experienced watching the Athenian political process.
Pyrrhic War. The balance of power in Europe swings westwards.
The ancient Greeks in general. In terms of literature, history, architecture, arts, politics their influence was enormous.