Before you default Frebaut, let's get clarification on the OP.
Do you mean "How many classes can we learn at once?" or "How many learned skills from these classes can we have active at once?"?
The way the OP is stated is ambiguous. A 'class' refers to "Carpenter" or "Botanist", for example...while a skill can refer to "Epiphany", "Maker's Muse", or "Preserve" - actions you can equip from reaching a certain threshold in a class.
So, to the OP - you can learn as many classes are as available in the game, simply by acquiring and equipping an appropriate arm or tool (Main Hand). The amount of skills you can have active at any given time scales with the rank of the active class... for example if you have a Rank 50 Goldsmith, but a Rank 1 Armorer, while Goldsmith you'll have more action slots available to equip more concurrent skills.