  We study the maximal operator on the variable exponent H\"older spaces in the
setting of metric measure spaces. The boundedness is proven for metric measure
spaces satisfying an annular decay property. Let us stress that there are no
assumptions on the regularity of the variable exponent and the variable
exponent can touch values $0$ and $1$. Furthermore, the continuity of the
maximal operator between H\"older spaces is investigated. Those results are new
even in the Euclidean setting.
