A Banach algebra A is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when A⁎=WAP(A). To identify the opposite behaviour, Granirer called a Banach algebra extremely non-Arens regular (enAr, for short) when the quotient A⁎/WAP(A) contains a closed subspace that has A⁎ as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra A is r-enAr, with r≥1, when there is an isomorphism with distortion r of A⁎ into A⁎/WAP(A). When r=1, we obtain an isometric isomorphism and we say that A is isometrically enAr. We then identify sufficient conditions for the predual V⁎ of a von Neumann algebra V to be r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr:(i)the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr,(ii)the weighted group algebra of any non-discrete locally compact infinite group and for any weight,(iii)the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr. The Fourier algebra A(G) of a locally compact infinite group G is shown to be isometrically enAr provided that (1) the local weight of G is greater or equal than its compact covering number, or (2) G is countable and contains an infinite amenable subgroup.