In most types of polymerization, the word "catalyst" has been replaced by the word "initiator". The exception is, e.g., coordination polymerization where we can talk about "Ziegler-Natta catalysts".
The growth of polymer occurs in both of the main methods of polymerization: chain-reaction polymerization & step-reaction polymerization.
As for chain-reaction polymerization, more amount of initiator will result in shorter polymeric chains (i.e. lower average molecular mass).
In step-reaction polymerizations, there is a different story. For these and case by case, you may use initiator (e.g. in grafting) or catalyst (e.g. in the production of PET by transesterification) but the advice of prominent scientists had been to use very little amount.