The Nordic Countries are very well bonded, with quite a few special agreements among them.
How about Belgium, the Netherlands, and Luxembourg, collectively known as Benelux?
Even before Russia took control of the Eastern Bloc nations, many Slavic lands (including Bulgaria and Serbia) regarded Russia as their big brother, as their friend who’d liberate and protect them from the Turks.
I don’t know if many STILL love Russia the same way, but proud Slavs from many nations have historically looked to Russia as their shining star, the nation who could unite all Slavs.
I think Poland once had a “special relationship” with France (though modern Frenchmen may see Poles as economic competitors, rather than as allies, these days).
Probably the best suggestion, although borne of necessity rather than political expedience, being a collection of small countries which exist (or continue to exist) through accidents of history rather than economic or military might. Much of the free-trade principles have been subsumed into EU-level co-operation, anyway.
And what makes it REALLY special is that the US replaced the UK as the dominant global naval power with nary a shot fired between the two. The replacement of one hegemon by another without some armed conflict between the two is, as far as I know, otherwise unprecedented.
Oops. Meant to start a new OP.
Consider this a bump?