  We establish several new probabilistic, dynamical, dimensional and number
theoretical phenomena connected with Ostrogradsky-Sierpi\'nski-Pierce
expansion.
  First of all, we develop metric, ergodic and dimensional theories of the
Ostrogradsky-Sierpi\'nski-Pierce expansion. In particular, it is proven that
for Lebesgue almost all real numbers any digit $i$ from the alphabet $A=
\mathbb{N} $ appears only finitely many times in the difference-version of the
Ostrogradsky-Sierpi\'nski-Pierce expansion.
  Properties of the symbolic dynamical system generated by a
shift-transformation $T$ on the difference-version of the
Ostrogradsky-Sierpi\'nski-Pierce expansion are also studied in details. It is
shown that there are no probability measures which are invariant and ergodic
(w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure).
  Thirdly, we study properties of random variables $\eta$ with independent
identically distributed differences of the Ostrogradsky-Sierpi\'nski-Pierce
expansion. Necessary and sufficient conditions for $\eta$ to be discrete resp.
singularly continuous are found. We prove that $\eta$ can not be absolutely
continuously distributed.
