  Several theories have been proposed to generalise the concept of analytic
continuation to holomorphic functions of the disc for which the circle is a
natural boundary. Elaborating on Breuer-Simon's work on "right limits" of power
series, Baladi-Marmi-Sauzin recently introduced the notion of "renascent right
limit" and "rrl-continuation". We discuss a few examples and consider
particularly the classical example of "Poincar{\'e} simple pole series" in this
light. These functions are represented in the disc as series of infinitely many
simple poles located on the circle; they appear for instance in small divisor
problems in dynamics. We prove that any such function admits a unique
rrl-continuation, which coincides with the function obtained outside the disc
by summing the simple pole expansion. We also discuss the relation with
monogenic regularity in the sense of Borel.
