  We consider the problem of the boundedness of maximal operators on BMO on
shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing
property, the corresponding maximal function is bounded from BMO to BLO,
generalising a known result of Bennett for the basis of cubes. When the basis
of shapes does not possess an engulfing property but exhibits a product
structure with respect to lower-dimensional shapes coming from bases that do
possess an engulfing property, we show that the corresponding maximal function
is bounded from BMO to a space we define and call rectangular BLO.
