Okay, here's one.
Athens fell. Large-scale immigration of Greeks to "Magna Grecia" seriously butterflied the rise of Rome, and what with peninsular Greeks, meddling Persians, Carthaginians, and, a century and a half after the fall of Athens, a breakway Satrapy of Greece, the Roman Empire never really got up to launch speed.
Time passed. Carthage built it's own empire in North Africa, Italy, Iberia, and S. Gaul, even grabbing Egypt for a while from a distracted Persia, but decayed in time. Germans invaded north Gaul, Gauls invaded Italy, and Huns invaded everyone, followed by Slavs. A revived Persian Empire burned Carthage to the ground. Scandinavians brought good government and many decapitations to the British Isles and the eastern Baltic. Persia grew and shrank, passed under a Turkish dynasty for a while, but never ceased to be entirely, as much a fixture of the landscape as some form of China in our world. New religions arose in the west, and new states in the east. The Carthaginian successor state in the west discovered a new world.
Currently, the world is dominated by the ancient Persian empire and by the powerful German-settled republic centered on what in our world would be the Mississippi. The Chinese, although battered by invasions from Turkic people deflected eastwards during periods of Persian strength, have reconsolidated to the south and are rapidly rising to make a third great power.
The North American Republic is rather oligarchic, but there is plenty of opportunity to rise in a highly competetive society. It speaks a German language influenced by Celtic and Punic, and follows, along with most of the world west of the Slavic and Persian lands, a religion drawing from the teachings of a celtic prophet from the Celtic-Greek borderland, also including elements of Judaism and Punic thought, and embroidered by centuries of imaginative recopying and at least two "reformations." The version followed by most inhabitants of Confederation is tolerant and easy-going enough, although there is a nasty streak to their otherwise artistic and sophisticated society (the death penalty is widely applied, and in a stroll through any large city one can usually find the chemically preserved body of an executed criminal on prominent display).
The Confederation is allied to several key European states to prevent any expansion of Persian power in the area, especially into backward eastern Europe. (Which is, oddly enough, Jewish. The great medieval Slavic empire adopted Judaism, like the Khazars of OTL, and it actually stuck. Religious hostility has meant for poor transmission of ideas, and the Slavic nations suffered badly in the last war, with the Gottslanders tearing a slab off Sklavonia and setting it up as a puppet and the Greco-Armenian empire actually annexing a chunk of Slavic territory outright).
Persia is a beaurocratic, technocratic dictatorship, with the Shah largely a figurehead nowadays. It follows a heavily mutated form of Zoroastrianism, and dominates much of India through puppet regimes. The Bod Empire and the Turks are strong enough to discourage any further expansion east, but the Empire works to stregnthen its position in south and SE Asia, Africa, and eastern Europe: in Africa, they have created a large client state organization in the Central Aethiopian Economic Community.
Its relationship to the Greco-Armenian empire is...complex. The area has come in and out of Persian control over the last two millenia, there have been intermarriages between the royal houses and Greek thought and philosophy have been terribly important in Persian intellectual history. OTOH, they follow a different branch of Zoroastrianism, and Greek pundits like to use the 'decadent' word a lot when talking about Persia.
OTL Latin America is a mix of former and current German/Gottslandish, British/Albien, Gallic and Carthaginian colonies. South America, under the more-or-less leadership of the carthaginian-derived New Tartesh', is essentially neutral where not part of a European power. The Quapac republic is the one state dominated by people of native American extraction, while to the north the Abenak Free State is fairly close to anarcho-capitalist.
Currently, the Persians, the Gettislanders, and the rising power of the Central Realm (sortof like our China, but with a Zoroastrian-influenced dualist philosophy, no Buddhism, and ideas of kingship also influenced by Persian models) are kissing and making up in the face of global warming worse than OTL: the southwest of the Gettisland Confederation is in the fifth year of a drought, and much of Persia is under water rationing - a lot of perfume is being used instead of bathing.
Bruce