Let $f$ be a holomorphic endomorphism of $\mathbb{P}^k(\C)$ 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 $\mathbb{P}^2$.