If you fix a particular axiom system for deriving your termination proofs (e.g. because you write them all down in Coq), then for Gödelian reasons, there are programs that don't terminate but for which a nontermination proof doesn't exist (and thus can never be found). If Coq is consistent, then the program that enumerates all possible Coq proofs and stops as soon as it finds a contradiction, is one such program.