Polyglutamylase which modifies tubulin, generating polyglutamate side chains of variable lengths on the gamma-carboxyl group of specific glutamate residues within the C-terminal tail of tubulin. Mediates both ATP-dependent initiation and elongation steps of the polyglutamylation reaction. Preferentially modifies the beta-tubulin tail over an alpha-tail. Competes with monoglycylase TTLL3 for modification site on beta-tubulin substrate, thereby creating an anticorrelation between glycylation and glutamylation reactions. Required for neurite growth; responsible for the strong increase in tubulin polyglutamylation during postnatal neuronal maturation.