Abstract It is well known that if the Hardy–Littlewood maximal operator is bounded in the variable exponent Lebesgue space L p ( ⋅ ) [ 0 ; 1 ] ${L^{p(\,\cdot\,)}[0;1]}$ , then p ( ⋅ ) ∈ BMO 1 / log ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$ . On the other hand, there exists an exponent p ( ⋅ ) ∈ BMO 1 / log ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$ , 1 < p - ≤ p + < ∞ ${1<p_{-}\leq p_{+}<\infty}$ , such that the Hardy–Littlewood maximal operator is not bounded in L p ( ⋅ ) [ 0 ; 1 ] ${L^{p(\,\cdot\,)}[0;1]}$ . But for any exponent p ( ⋅ ) ∈ BMO 1 / log ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$ , 1 < p - ≤ p + < ∞ ${1<p_{-}\leq p_{+}<\infty}$ , there exists a constant c > 0 ${c>0}$ such that the Hardy–Littlewood maximal operator is bounded in L p ( ⋅ ) + c [ 0 ; 1 ] ${L^{p(\,\cdot\,)+c}[0;1]}$ . In this paper, we construct an exponent p ( ⋅ ) ${p(\,\cdot\,)}$ , 1 < p - ≤ p + < ∞ ${1<p_{-}\leq p_{+}<\infty}$ , 1 / p ( ⋅ ) ∈ BLO 1 / log ${1/p(\,\cdot\,)\in\mathrm{BLO}^{1/\log}}$ such that the Hardy–Littlewood maximal operator is not bounded in L p ( ⋅ ) [ 0 ; 1 ] ${L^{p(\,\cdot\,)}[0;1]}$ .