  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 the Euler constant is an exponential
integrated algebraic number. Hence they are a good candiate for a natural
number system extending the period ring and containing important mathematical
constants.
