Plausibility Question: A Larger Achaemenid Empire

Does anyone have any ideas on how much larger the Achaemenid Empire could have gotten?

For instance, I had previously asked in another thread about the possibility of conquering Carthage.
 
Not sure. Maybe parts of India?
Or conquering down the Nile, and taking on Axumite region?
Making the Caspian Sea their version of a 'Mare Nostrum' (or whatever the term would be in their language)?
Conquering Greece, and then taking on Italy?
 
I think maximum Persia would be Greece, Thrace, maybe Macedonia, and Carthage. I doubt the Achaemenids would be able to hold that land for a very long time. But if they were able to, I could see the regions being somewhat Persianized after a while.

Things would almost immediately get interesting (and unrecognizable) from then on out.

Much larger Persian cultural sphere?
Maybe the Romans are butterflied away...
Possibly a larger Greek immigration to Massalia (Marseilles), Hellenistic Gaul?
 
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