Catalyzes the trimethylation of eukaryotic elongation factor 2 (EEF2) on 'Lys-525'. L-lysyl-[protein] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl-[protein] + 3 S-adenosyl-L-homocysteine Interacts with FAM86B2 and FAM86C1P. Belongs to the class I-like SAM-binding methyltransferase superfamily. EEF2KMT family.