Alexander III of Macedon is referred to as Alexander "the great", but who gave him this title? At what point in history was he acknowledged as such and why is he considered so great?
I think the first time he is mentioned as "Alexander the Great" (at least in the sources known to us) is Quintus Curtius Rufus' "Historiae Alexandri Magni Macedonis", this "Magni" has been translated into English as "Great". Here it clearly refers to his talent as a military leader which allowed him to build up a huge empire. Quintus Curtius Rufus was a Roman historian, this book is considered to be written in the 1st century AD.
Btw, Russian historians tended to focus on the word "Macedonis" which is why he is better known as Александр Македонский (Alexander of Macedon) in Russian.
Alexander apparently received that epithet from the Romans, who admired him. The oldest surviving reference of the title is found in the Mostellaria ("The Haunted House"), a play written by Titus Maccius Plautus (c. 254 – 184 BC). This roughly a century or so after Alexander's death in 323 BC.
Tranio: Alexandrum magnum, atque Agathoclem, aiunt maxumas Duo res gessisse. Quid mihi fiet tertio, Qui solus facio facinora immortalia?` - Mostellaria. III. 2. Titus Maccius Plautus.
Plautus seems to assume his audience would recognise who
Alexandrum magnum was, an indication that the epithet may have predated his Mostellaria. It appears the Romans began styling Alexander as "the Great" shortly (relatively speaking) after his death.
The reasons are of course obvious, as @TomAu and @WladimirPalant have explained.
It took a long time before Alexander's greatness really came home to the Romans, although they were the people who, as far as we can tell, gave him the epithet 'the Great', and it took even longer before he made an impact on individual Romans. The earliest record we have of Alexander is in Plautus' Mostellaria, 775-7, where the slave Tranio brackets him with Agathocles, tyrant of Syracuse: Alexandrum magnum atque Agathoclem aiunt maxumas
- Den Hengst, Danièel. Emperors and Historiography: Collected Essays on the Literature of the Roman Empire by Danièel Den Hengst. Vol. 319. Brill, 2010.
Alexander was called "the Great" by historians shortly after his death in his early 30s.
During his short life, we conquered essentially all of the modern Middle East and Egypt, starting with a base of Macedonia, a kingdom near Greece. In the process, he defeated much larger, mostly Persian armies, in an unbroken series of battlefield victories and successfully campaigns that are studied even to this day. Had he lived longer, he might have united large parts of Europe and Asia under Macedonian rule.
Despite his bellicosity, his rule was mainly benevolent. He treated conquered peoples kindly (relative to his time), and encouraged his Greeks to intermarry on equal terms with these people. Thus, Greek civilization spread throughout much of the Middle East with little further bloodshed. He also supported the development of learning, and his tutor, Aristotle, plus earlier scholars such as Socrates and Plato, were passed along to later generations as members the great minds of antiquity.
Citations, while useful, are not allways necessary. Suffice it to observe that he did things no one achieved before, in scale, rapidity, modernity and efficiency: 1. Created the largest sustainable empire by size as of then. 2. Conquered and settled in lands never conquered before by the Greeks (the whole of Mesopotamia and the Persian Empire, Afghanistan). 2bis. Won even when outnumbered by a significant ratio. 3. Waged war further than any other greek general (India). 4. Ended the last native Pharaonic dynasty (to replace it by the dynasty of one of his generals: Ptolemy I Soter). 5. Built or renamed at least 13 cities after his name (still named after him today: Kandahar (Afghanistan); Alexandria (Egypt)).. 6. All this before the age of 32. 7. Was considered a local divinity in various parts of his empire. 8. Never came back home, even for a day.
While sources of whether he really did that are necessary. Sources quoting him 'the Great' are pointless. If he did this, he was indeed great in many instances as compared with other mortals .