In the 1950s, the government of the Soviet Union (I think it was Khruschev) offered the Kaliningrad Oblast to the government of the Lithuanian SSR. At the time, it was seen by the Soviets as a useless piece of land, an enclave with nothing of value.
However, the LSSR government refused.
So what would've happened if they accepted, then?
I had a soc.history.what-if post about this some time ago
https://groups.google.com/d/msg/soc.history.what-if/Mng2mbFotzs/JtrJ3-wDd9YJ where I expressed suspicion that Lithuanian First Secretary Snieckus might have encouraged this story to boost his popularity among nationally-minded Lithuanians:
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Interestingly, there were rumors that *Stalin* had offered Snieckus the same thing and met with the same refusal:
"In the 1960s and '70s, rumor circulated in Lithuania that in 1945 Stalin had offered the region to Lithuania but that the then First Secretary, Antanas Snieckus, had adroitly managed to refuse. Lithuania possessed neither the manpower nor the resources to absorb and reconstruct the territory, and if it had been incorporated into the LiSSR, the latter's population would have been less than 60 percent Lithuanian rather than around 80 percent, which it has remained since the early 1960s. Cynics have interpreted such rumors as a clever move to defuse a potentially troublesome question while simultaneously raising the stature of the long-serving First Secretary in the eyes of nationally-inclined Lithuanians. Whether there is any factual basis for such conjectures cannot be ascertained, but their currency underscores the significance of the question in Lithuania. Romuald J. Misiunas and Rein Taagepera, *The Baltic States, Years of Dependence, 1940-1990*, p. 347.
http://books.google.com/books?id=vrrBLJtDXb4C&pg=PA347
The reports with regard to Khrushchev seem a bit more plausible. First of all, it's a bit hard seeing the party secretary of a Union Republic saying No to Stalin. As I note at
http://groups.google.com/group/soc.history.what-if/msg/a5dd5f01f53ebd4d "Of course, Snieckus would never dare to openly object to an influx of ethnic Russians into Lithuania. What he *might* conceivably have said is something like 'Comrade Stalin, the Lithuanian working people are deeply honored by your offer of northern East Prussia, but we believe that since the great Russian people have more than any other borne the brunt of this war, they should be the ones to get this territory.' But I doubt that he even said that." It's at least a little more plausible seeing Snieckus say No to Khrushchev, at least if Khrushchev only offered it as a suggestion. Second, as Misiunas and Taagepera note, it did seem plausible that the Khrushchev reorganizations (*sovnarkhozy* or regional economic councils) would lead to the oblast being attached to Lithuania. "In the spring of 1957, a suggestion appeared that the Couronian Bay should entirely be incorporated into Lithuania, and in 1963, the management of its industry was turned over to Lithuania; its railroads and inland waterways had been under Lithuanian administration for many years. Agriculture, however, was not at that time attached to the Lithuanian sovnarkhoz, and it remains unclear whether any of these arrangements with Lithuania would have survived the abolition of the sovnarkhozy in 1965 and if so in what form..."
http://books.google.com/books?id=vrrBLJtDXb4C&pg=PA347