At this time period, I would say that the Ottoman Empire would be the most likely answer for a few reasons. I am making some assumptions from your question however, chief among them that despite Ottoman gains in Eastern Europe, the Turks were seen as an Asian 'other' with a false religion, and so they would be considered non-European despite being neighbours geographically.

1557 takes us nine years before the death of the sultan Suleiman the Magnificent, who perished due to an injury while campaigning in Hungary. Suleiman was well known to Western Europeans due to his campaigns in Eastern Europe and the Mediterranean. Suleiman's forces took Belgrade, Buda, Rhodes, and having won many battles (though crucially never being able to conquer Vienna), Suleiman's military accomplishments made him notorious in the Christian world. He also viewed himself as an equal, if not superior, to the Holy Roman Emperor Charles V; it was common for sultans since Mehmed II to cite their conquest of Constantinople, the former seat of the Byzantines, as proof that they were the new Caesars. Europeans would also be aware of the architectural achievements Suleiman oversaw: two magnificent mosques were built in Constantinople, and holy sites in Jerusalem were renovated and maintained.

Suleiman is known for many more achievements by historians, but the issues I listed above would have been the most likely facts that educated Europeans knew about the Ottomans. I do note that I'm conflating 'high' civilisation with 'powerful' and/or 'influential'. Places such as China and Persia would have been known to Europeans, but I think that the Ottoman's proximity would have lead them to be at the front of Western European's minds.

Sources: Caroline Finkel, "Osman's Dream: The History of the Ottoman Empire" Karen Barkey, "Empire of DIfference: The Ottomans in Comparative Perspective"

Do you mean a contemporary society to themselves, or can they include past societies?

One alternative to /u/snerdsnerd's proposal of the Ottoman Empire that I can think of is Persia under Shah Abbas, who reigned slightly after the date of your question (r. 1588-1629.)

Shah Abbas launched the first Persian embassies to Europe and European guests of the Shah like Robert Shirley experienced the full grandeur of his court and of his magnificent capital, Isfahan. In a general sense Persia under Shah Abbas reached an extraordinary level of refinement in art and culture, but from a European perspective it would also have been somewhat familiar given the trade and political contacts between Europe and Persia that emerged at this time.