For p ∈ [ 1 , ∞ ) p\in [1,\infty ) , we prove that simple, separable, monotracial UHF L p L^{p} -operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF L p L^{p} -operator algebras of tensor product type obtained from a diagonal system of similarities. For p = 2 p=2 , it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and particularly Hjorth’s theory of turbulence.