The question of maximum number never really arose, but it's remarkable that we never saw more than six people beamed at a time. Indeed, in "The Apple", it appears that at first, six people beam down, and only then do three more appear - and no explicit story logic is provided for why the latter three (McCoy and two guards) would have opted to be delayed.
On the other hand, the number of pads on the transporter platform doesn't seem to be closely related to what is being transported, considering that people and objects may extend past the limits of a pad or cover several of them (say, when lying down injured). Which makes for an odd contrast with "Where No Man Has Gone Before" where the unconscious Gary Mitchell is deliberately held standing upright within the supposed "confines" of a single pad, and our other two characters pull their supporting hands away for the duration of the transport! Such a weird maneuver would seem to require pretty pressing reasons, which are never given.
On a separate issue, the ship probably had several of those six-pad transporters, as the single set underwent modifications that more or less force us to assume it represented several separate spaces. Moreover, the transporter was indicated to be on different parts of the ship in different episodes - for example in "Dagger of the Mind" it brings van Gelder to the lower levels rather than to the more familiar saucer Deck Seven. But perhaps only one platform could send its signal down to the target planet at a time, explaining "The Apple"? Certainly some
transporter resources were bottlenecks and single points of failure, as seen in "The Enemy Within"...