Max Sinister said:
Russia in OTL called itself the 3rd Rome. WI they go a step further and form a union with Byzantium? Over the centuries, the Russians adapt Byzantine ways: At first, the court starts speaking Greek, then all the nobles, then the church later the middle class, at the end, everyone.
There's your Greek superpower
Not sure how viable it is; in OTL in early XIXth century most Russian nobles' first language was French, which was considered more prestigious than Russian, however, as Russian nationalism garnered strength, it mostly went away.
Now there is another alternative - Russia did not start calling itself Third Rome until at least XVIth century, if not later. Let's say, Byzantium manages to somehow survive for another hundred years (although it would be hard, given that the Ottomans really did want the city - maybe if somehow they decide to let the city be their vassal and leave it in peace, but this might require some ASBs), and through dynastic ties enters into a personal union with Russia... if Ivan the Terrible is still born, he could inherit Byzantine throne if there are no other male descendants of Paleologi out there that have a better claim. He would definitely be a somewhat legitimate ruler, and could reinforce Constantinople enough to where it would not fall, and would be able to begin reconquest of Greece and Smyrna.
Another alternative - Catherine the Great was getting rather close to reform Byzantium with one of her grandsons as the new Emperor. What if she succeeded? From what I recall, she had said grandson (Constantine) raised in a way that would make him more appealing to the Greeks (it is said he was taught Greek language before Russian), and practically tried to groom him for the Byzantine crown. In this case, we can have Russian-backed neo-Byzantium in XVIIIth century, that over time grows to be a superpower as with Russian backing it slowly but surely dismantles the Ottomans.
Through some butterflies, there can be earlier Italian unification under the mandate of the French crown, or if there is still Napoleon, maybe one of his family members (or even Nappy himself) ends up in control of Italy... if Corsica does not become French possession, or France loses it before Nappy is born (if he is still born, that is), Nappy could be the instrument of Italian unification, and a founder of Italian dynasty that would have just enough power at first to keep itself independent and even get a colony or two here and there, but not enough to wage European war.
Fast forward a century, by now we are getting close to 1900. Neo-Byzantines are by now in control of large parts of Africa as colonies, and the Italians are looking that way themselves, having been a united nation for much longer, and having grown in strength. Both have the aspirations of "new Rome", and both are expansionistic and aggressive; by now Russia has enough problems of its own to let Byzantium do its own thing, although they are still allied.
Not sure if WWI still would occur, as chances are it is butterflied away, but if there is something akin to it, there could be Russian bloc (which will include Byzantium) on one side, possibly including France and maybe Britain, and another bloc, maybe Germany, Austria, and Italy, who is jealously looking at the Byzantine African protectorates and colonies as the means to set up its own empire. If the war does happen, say things happen similar to OTL, the Central Powers getting the worst of things, Russia proper still suffering a revolution, and whatever is left of the Romanovs packing up to Constantinople (where another branch of the family is still enthroned). However, here Italy gets the worst of it, as opposed to Germany, in the eventual version of TTL's Versailles treaty, giving rise to something awkwardly similar to fascism, spiced up with the dreams of old Roman power, and xenophobic hatred of the Byzantines, who in Italian eyes are the "usurpers" of the Roman mantle.
So, a Hitler-analogue rises in Italy, however, he is less bent on conquest than on colonization, and after a few decades gets his way in North Africa and wherever else... since he is less of a wacko than OTL Adolph, he knows fully well that another World War would be suicidal. In a meanwhile, the Byzantines seize as much of the Middle East as they possibly can, getting to oil there, and creating lots of trouble with the Muslim locals, and ending up with a guerilla war on their hands.
With oil revenues, however, the Byzzies get richer and richer with every day, and with an eventual technocrat of sorts in power in Italy, military buildup across both sides of the Adriatic begins - think Cold War. Some time by 1950s, someone comes up with the idea of the nukes, and both sides hungrily arm themselves with anything they can get their hands on. Come 2005, and two nuclear powers still look warily at each other's direction, even though relations had somewhat warmed in the recent decades.
OK, it requires some luck and whatnot, but IMO the above scenario at least has moderate plausibility.