ra.rings and algebras - What is an exponential?

Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on

• tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential map sends a vector to a point on a curve naturally defined in terms of the vector;


• unital Banach algebras?

(NB. I am not conversant with category theory beyond the words "morphism" and "functor". But a categorically flavored answer that takes my limited knowledge base into account would be preferable. An internet search led me to the notion of a "Cartesian closed category", which doesn't seem to be the sort of thing I have in mind.)