Just some… sidenote rant:
ανοιχτοχέρης […] is used more in the sense of a spendthrift.
+1
As for
μεγαλόδωρος, in my ears it sounds like something you might a hear a politician say as a way of paying tribute to some senior public official or whatever…
Δοτικός, on the other hand, as cougr already implied, has more abstract connotations – which, admittedly though,
are present in the word of your query (
γενναιόδωρος), as is the case with the word
generous, I guess. (You could almost say that those two are the hypernyms of the semantic field in question…) But
δοτικός typically describes someone who's being generous in terms of emotional support, hospitality, and less often in terms of money. And if we go that far, we might as well include
ανοιχτόκαρδος in the list.
Λαρτζ is spot-on, and wiiidely used – which is somewhat hilarious actually, since the usage of the English word
large in this sense is more or less obsolete, if I'm not mistaken.
Depending on where you hang around, you might also come across the derivative noun
λαρτζιλίκι (plural:
λαρτζιλίκια).
Another funny one I came to think of, is something my grandpa used to say (and my dad may still ocassionally recite) when someone would pay the bill at a restaurant or something:
κάνει/έκανε τον Αμερικάνο!
Cheers!