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 (PubMed:16901895, PubMed:25959773). Mediates both ATP-dependent initiation and elongation steps of the polyglutamylation reaction (PubMed:16901895, PubMed:25959773). Preferentially modifies the beta- tubulin tail over an alpha-tail (PubMed:16901895, PubMed:25959773). Competes with monoglycylase TTLL3 for modification site on beta-tubulin substrate, thereby creating an anticorrelation between glycylation and glutamylation reactions (By similarity). Required for neurite growth; responsible for the strong increase in tubulin polyglutamylation during postnatal neuronal maturation (By similarity). {ECO:0000250|UniProtKB:A4Q9F0, ECO:0000250|UniProtKB:F7E540, ECO:0000269|PubMed:16901895, ECO:0000269|PubMed:25959773}. This is the function of ENSG00000137941 (TTLL7, tubulin tyrosine ligase like 7).