Periods are defined as integrals of semialgebraic functions defined over the rationals.

Periods form a countable ring not much is known about. Examples are given by

taking the antiderivative of a power series which is algebraic over the polynomial ring over

the rationals and evaluate it at a rational number. We follow this path and close these algebraic

power series under taking iterated antiderivatives and nearby algebraic and geometric

operations. We obtain a system of rings of power series whose coefficients form a countable

real closed field. Using techniques from o-minimality we are able to show that every period

belongs to this field. In the setting of o-minimality we define exponential integrated algebraic

numbers and show that exponential periods and the Euler constant are exponential

integrated algebraic number. Hence they are a good candiate for a natural number system

extending the period ring and containing important mathematical constants.