  This paper is inspired by a class of infinite order differential operators
arising in the time evolution of superoscillations. Recently, infinite order
differential operators have been considered and characterized on the spaces of
entire monogenic functions, i.e., functions that are in the kernel of the Dirac
operators. The focus of this paper is the characterization of infinite order
differential operators that act continuously on a different class of
hyperholomorphic functions, called slice hyperholomorphic functions with values
in a Clifford algebra. We introduce the concept of proximate order and
establish some fundamental properties of entire hyperholomorphic functions that
are crucial for this characterization.
