The correct IUPAC name of $[\mathrm{Co}(\mathrm{NH}{}_3){}_6]\mathrm{Cl}{}_3$ is

NH3 is named ammine (two m) as a ligand. Charge balance: three $\mathrm{Cl}{}^{-}$ balance the cation, so $x+0=+3$ and Co is +3. Counter-ions are named without a multiplying prefix, so the name is hexaamminecobalt(III) chloride.