One of the most consistent and universal features of Greek warfare is the idea that a commander must lead from the front. This is how he proves his worth as a leader and inspires his men to follow his example. Already in the earliest epics, lords know they must be first at the danger if they are to have the respect of their men:

'It becomes us, therefore, to take our stand in front of all the Lycians and bear the brunt of the fight, so that one may say to another, "our lords in Lycia eat the fat of the land and drink the sweetest wine, but they are good men; they fight well and are ever at the front in battle."'

-- Homer, Iliad 12.316-321

In battle, these lords take on the role of promachoi, literally "front-fighters" - those whose heavy armour allowed them to step out ahead of the mass of warriors to challenge the enemy directly. The Spartan poet Tyrtaios (mid-7th century BC) describes the ethos of the promachoi:

Do not fear throngs of men or run in flight, but let a man hold his shield straight toward the front, despising life and loving the black death-spirits no less than the rays of the sun. You know how destructive the deeds of woeful Ares are; you have learned well the nature of grim war, you have been with the pursuers and the pursued, you young men, and you have had more than your fill of both. Those who dare to stand fast at one another’s side and to advance towards the front in hand-to-hand conflict, they die in fewer numbers, and they keep safe the troops behind them; but when men run away, all excellence is lost.

-- Tyrtaios fr. 11, 3-14

As Greek infantry combat became more organised, the promachoi gradually merged into regular formations, and battles no longer offered much scope for individual bravery. Nevertheless, it remained a principle that the generals of Greek armies fought in the front rank, and that this front rank ought to consist of those men who were most willing and able to fight. Since hoplite formations consisted of untrained amateurs, inspiring examples were of critical importance; a phalanx wouldn't function very well in battle if its leader refused to show his men what was expected of them. Xenophon sums up the valour of his friend, the Spartan king Agesilaos, simply like this:

As for courage, he seems to me to have afforded clear proofs of that by always engaging himself to fight against the strongest enemies of his state and of Greece, and by always placing himself in front in these fights.

-- Xenophon, Agesilaos 6.1

This Athenian author regarded the general's example in battle as an essential element of leading armies. He has Sokrates point out to a young man who wanted to learn the arts of generalship that "under all conditions, human beings are most willing to obey those they believe to be the best" (Memorabilia 3.3.9). In battle, this means those who are willing to take the lead, show unflinching bravery and self-control, and take the fight to the enemy.

The clearest evidence that Greek generals literally lived and died by this principle is the sheer number of them who are reported to have died fighting in the front ranks of their armies. At Marathon, the earliest historical battle of which a full description survives, the Athenian commander Kallimachos was killed, along with Stesilaos, another member of the board of generals (Herodotos 6.114). At Spartolos in 429 BC, the defeated Athenians are said to have lost 400 men 'and all their generals' (Thucydides 2.79.7). At the battle of Amphipolis in 422 BC, neither the defeated Athenian Kleon nor the victorious Spartan Brasidas survived the fighting. The famous Theban general Epameinondas fell in the front ranks at Mantineia in 362 BC, just 3 years after his colleague Pelopidas had met the same fate:

Through the rear ranks he forced his way to the front, and filled all his men with such vigour and ardour that the enemy also thought them changed men, advancing to the attack with other bodies and spirits. (...) Then Pelopidas, looking down from the heights and seeing that the whole army of the enemy, though not yet put to flight, was already becoming full of tumult and confusion, stood and looked about him in search of Alexander [the tyrant of Pherai]. And when he saw him on the right wing, marshalling and encouraging his mercenaries, he could not subject his anger to his judgement, but, inflamed at the sight, and surrendering himself and his conduct of the enterprise to his passion, he sprang out far in front of the rest and rushed with challenging cries upon the tyrant. He, however, did not receive nor await the onset, but fled back to his guards and hid himself among them. The foremost of the mercenaries, coming to close quarters with Pelopidas, were beaten back by him; some also were struck and killed; but most of them fought at longer range, thrusting their spears through his armour and covering him with wounds.

-- Plutarch, Life of Pelopidas 32.4-7

This was the tradition into which Alexander the Great was born. He was a great admirer of Homer's epics, he knew all about the exploits of Xenophon and the Ten Thousand in Persia, and his father had spent his formative years as a hostage in Thebes learning the principles of generalship from Epameinondas. In short, he was infused with the ethos that generals must lead from the front, that the morale of fighting men hinged on giving them the right example, and that there was nothing more glorious than falling in the fury of battle. Add to this the fact that rulership in Macedon was propped up by very personal ties of loyalty and camaraderie between members of the elite, and it becomes obvious that Alexander was never going to shy away from danger in battle. He had to prove himself worthy to lead, and worthy to be followed. He had to show his men what Xenophon regarded as the fundamental truth of military leadership:

To put it shortly, a commander is least likely to incur the contempt of his men if he shows himself more capable than they of doing whatever he requires of them.

-- Xenophon, The Cavalry Commander 6.4

It is undeniable that Alexander frequently risked his life in close combat. We know he was repeatedly wounded in battle and narrowly escaped death on several occasions. Any of his major engagements could have been his last; it is to some extent a miracle that he managed to die in bed and at peace. But had he lived longer, there is no doubt that he would have continued to display his reckless bravery in the front ranks, being first upon the enemy in his cavalry charges and siege assaults alike. And his biographers would have been able to share even more scenes like the one you heard about:

For after dispersing the inhabitants from the walls with missiles, he was the first to mount upon the wall by a scaling ladder, and since the ladder was broken to pieces and he was exposed to the missiles of the barbarians who stood along the wall below, almost alone as he was, he crouched and threw himself into the midst of the enemy, and by good fortune alighted on his feet. Then, as he brandished his arms, the barbarians thought that a shape of gleaming fire played in front of his person. Therefore at first they scattered and fled; but when they saw that he was accompanied by only two of his guards, they ran upon him, and some tried to wound him by thrusting their swords and spears through his armour as he defended himself, while one, standing a little further off, shot an arrow at him with such accuracy and force that it cut its way through his breastplate and fastened itself in his ribs at the breast. Such was the force of the blow that Alexander recoiled and sank to his knees, whereupon his assailant ran at him with drawn sword, while Peukestas and Limnaios defended him. Both of them were wounded, and Limnaios was killed; but Peukestas held out, and at last Alexander killed the barbarian.

-- Plutarch, Life of Alexander 63.2-4