Let \(M_n(\varepsilon)\) (for \(n\in \mathbb N\) and for \(\varepsilon >0\)) denote the set of \(x\in \mathbb R\) such that the inequality \[ |P(x)|<\prod_{1\leq i\leq m}\max(1,|a_i|)^{-1-\varepsilon} \] has infinitely many solutions \(P\in \mathbb Z[X]\) with deg \(P\leq n\) (these points are said to be very well multiplicatively approximable). This set is of measure zero (this result conjectured by A. Baker has been proved by D. Kleinbock and G. Margulis [Ann. Math. (2) 148, 339-360 (1988; Zbl 0922.11061)]. The aim of the paper is to prove that the Hausdorff dimension of the set \(M_n(\varepsilon)\) is larger than or equal to \(\frac 2{2+\varepsilon}\), and equals this value for \(n=2\). Furthermore, this number is conjectured to be the exact value of the dimension.