My documented maximum is 17 children borne by one woman (twice in my tree and in both cases involving no twins). For one of these (interestingly from the 16th century) I have names and baptismal dates for all 17 so I'm as confident as you can be that far back that this is right. The other is from the stated number of children born alive in the 1911 census, but I can only undisputably find 11. The surname is a common one, so I can't find children who were born and died between censuses without spending a sum the size of the national debt on certificates!
However, there is a statement in Burke's Peerage for one of my lines that one of my relatives had 20 children. I know of 14 and it is biologically possible for the remaining six to be interspersed between these 14 and to have died young, but from previous experience I know that BP often includes rather random facts that seem to have their basis in rumour rather than fact.
My GG grandfather was one of 15, including three sets of twins.