Under no criteria does Alexander's short-lived empire enter this realm of 'greatness'.
His empire was in ruins when he got it, he ruined it some more with his incessant warring, forced social upheaval by forced marriages between natives and Greeks and is neither the economic or military apex entity of its time.
I differentiate between modern history and pre-modern history because the realities of the modern world are starkly different than pre-modern world, since heavy machinery & 'lightning speed communciations' fundamentally change the human living condition and thus, social outlook.
However, there isn't much to differentiate the lives of common folks from the Frankish Empire to that of the Roman Empire and the Han, in the realms of technological and logistical game-changers.
This is why i nominate a modern and pre-modern candidates.
Ultimately, what makes a great empire is not just military might(though it is important), it is also about the influence of the society on subsequent generations and the quality of life.
In the former category, the Mauryas are as influential as the Romans, if not more: they exhibit religious tolerance ( which is absent from the Roman empire- in their pre-Christian days, they persecuted Christians and Jews, in their post Christian days, they persecuted pagans). The Mauryan empire, as most pre-Islamic Indian empires, patronized Buddhism, Jainism, Ajivika and Vedism (which developed into modern day hinduism), with individual emperors having their own personal creed ( Chandragupta Maurya was an ajivika follower, so was Bindusara, Ashoka was a Buddhist, his successor Dasaratha was a Buddhist, his successor Samprati was a Jain and his successor Salisuka was an Ajivak). This is remarkable for its time, where no empire, with the exception of the Achaemenid Empire & Maurya/Nanda Empires, officially endorsed different faiths.
The influence of the Mauryas is further maginified by the fact that Ashoka was the 'Constantine of Buddhism', without him, Buddhism would've been a minor following in the world, if even alive. Because of him, Buddhism spread through Central Asia and the Indian Ocean region and has come to influence the lives of over half a billion people today.
So clearly, the Mauryans tick the 'influence on subsequent generations' category pretty well.
On military might- i don't think there was an entity in its time that could challenge the Mauryas on power: the Selucids were the apex empire in the western world for much of the Mauryan time and it was a non-factor against the Mauryas: losing almost a third of its lands to the mauryas in a single campaign, not to mention, a large portion of the Selucid power was shored up by Mauryan support: Mauryan support gave Selucus 500 war elephants,which were instrumental in establishing Selucid supremacy in the Diadochi world, much later, similar aid was provided to Antiochus the Great after his failed Bactrian campaign.
Clearly, they tick the category of military and diplomatic influence categories pretty well too.
On social terms, they were the free-est society of their time: the only society to formally abolish slavery and garantee individual rights of both genders. In this respect, they make the Greeks and the Romans look barbaric in their social outlook. I rate the Indian society till the fall of the Gupta Empire ( 500s CE) to be the most developed and foreward thinking social entity in the world, as they were the only society to show modern day values such as seperation of church and state ( the Indian states sponsored multi-denominational faiths, unheard of till modern times- i am not talking about simply 'tolerating' them, but spending state funds on faith-related buildings on multiple denominations), fundamental rights of the individual ( which was lacking in the Greco-Roman world- their 'fundamental rights' were similar to the 'rights of the nobility' in the medeival world, where most 'rights' pertained only to aristoracy and the 'citizens aristocracy') and such modern concepts as protected species, animal rights, protected forests & national parks.
Given that they had public funded education and medicare ( the two are not unique to this civilization but still rather rare in the pre-modern world), i would say that the Mauryans and subsequent Indian dynasties were the most socially progressive in the world.
The caste system was not rigid, implying that its negatives were not felt in that society( caste becomes nothing more than an identity marker if inter-caste marriages are taking place and the son of a farmer could become a warrior, who's son could become a priest).
And lastly economy: the Indian economy of the Mauryan empire was the largest economy of its time, consistently showing a strikingly well developed industrial category ( Indian steel is the best steel of the ancient world, they were also the apex producer of textiles, fiber and dyes) alongside the world's most developed resource industry ( the spice route, till late 1st millenia BCE, was predominantly Indian goods shipped to the west).
Ultimately, if i wanted to be a commoner in the ancient world, i would pick the Mauryan society, where i would enjoy greater rights ( and especially, rights against being abused by previledged nobility- something that didn't exist for slaves in the Greco-Roman world) and social mobility. If i were to be a nobleman, i would also pick the Mauryan society-for it was the richest and most powerful of its time ( the two categories the nobles care the most about).
