Looking at the shop, you can purchase up to 12 slots, giving you 18 slots in total. Enough to try every available class if you wanted to. The character selection screen only has two tabs however.
Is there something missing here on how to open up a third tab? Can people buy slots 13-18 but not be able to use them? Using that many would be crazy, but I know myself and some others wanted to save slots for the class releases.