how many max games for non-payers?

how many max games does a non-payer get?

You can have a total of six games in progress at one time.

does it go up to 20 eventually as you continue to finish games or does it stay at six?

You can play as many games as you want here, but can only have six in progress at any one time.

During your first 20 games you will have a 'provisional' rating.