Universal source coding is, roughly, optimal compression without knowing the source but knowing a class it belongs to. An even more stringent property is adaptivity, performing well without even knowing the class, among a huge family of classes. Even mere universality is not possible without restrictions, and one typically assumes a finite alphabet with a known size. To model large alphabets, we instead propose classes with countably infinite alphabets characterized by a common dominating envelope. We give an explicit, computationally efficient, code that operates without knowing the envelope. It mixture-codes frequent symbols and integer-codes rare ones. Using regular variation theory and concentration of measure, we show that this code is near-adaptive to max-stable envelope classes.