  Let $f$ be a holomorphic endomorphism of $\mathbb{CP}^k$ having an attracting
set $A$. In this paper, we address the question of the "size" of $A$ in a
pluripolar sense. We introduce a conceptually simple framework to have
non-algebraic attracting sets. We prove that adding a dimensional condition,
these sets support a closed positive current with bounded quasi-potential
(which answers a question from T.C. Dinh). Therefore, they are not pluripolar.
Moreover, the examples are abundant on $\mathbb{CP}^k$.
