Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind which - by $M$. More precisely, for a function $f$ on $X$, define $$ Mf(x) := \sup_{B \ni x} \frac{1}{\mu(B)} \int_B f(y) \; dy,$$ with the supremum taken over either all balls centred at $x$ or over all balls containing $x$.

There are plenty of theorems of the form 'suppose $(X,d,\mu)$ has property A. Then $M$ is of weak-type (1,1)', (for example, property $A$ could be '$(X,d,\mu)$ is doubling') thus implying that $M$ is bounded on $L^p(X)$ for $1 < p \leq \infty$ (since $L^\infty$ boundedness of $M$ is always obvious).

I'm interested in the converse. What is an example of a space $(X,d,\mu)$ is *not* of weak-type (1,1)? And how would one go about proving such a result?

Further, if there are any spaces with $M$ not of weak-type (1,1), do we still have $L^p$ boundedness for any $p > 1$?