That's funny. There is in fact a quasi-finitist number scheme called the counting numbers, closed under +1 and addition/multiplication generally but not exponen

That's funny. There is in fact a quasi-finitist number scheme called the counting numbers, closed under +1 and addition/multiplication generally but not exponentiation, so unbounded but not too "problematic" as they have limited ability to represent propositions (or machine states, etc) arithmetically.