Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition.
What's a reference for that?
I don't have a complete reference (and like Tyler, I don't know exactly what result you want). But here are some observations:
[Added:] These two facts taken together imply that a "transfinite composition" of weak equivalences is a weak equivalence.
This answer serves to record two explicit proofs of this fact in the literature:
Corollary 5.1 in Raptis and Rosický, “The accessibility rank of weak equivalences”, arXiv:1403.3042v2. Theory and Applications of Categories 30:19 (2015), 687—703.
Theorem 4.6 in Barnea and Schlank, “Model structures on Ind categories and the accessibility rank of weak equivalences”, arXiv:1407.1817v6.