Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure.

Sorry if this has nothing to do, but from how I understood the question, at least this should be useful.

EDIT: Re-reading the question, I've noticed that you are maybe more concerned about balls. For this, it depends on what you want. For example, if you want, for any $\varepsilon$ a ball of smaller radius that has iterates with volume close to the manifold, it is easy to get (say, for example a north-south map). However, if you look for every ball at a time, this won't be true in general, for, example, if $f$ is not transitive, you get a reppeling neighborhood, so you won't aproach the total volume never. If $f$ is transitive, I believe you can manage to prove the same (that some balls won't increase their volume arbitrarily close to that of the manifold) but I have not an argument at the moment (it is clear if it is volume preserving, and otherwise, you will get some regions which are kind of ``repelling'').

Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure.

Sorry if this has nothing to do, but from how I understood the question, at least this should be useful.

EDIT: Re-reading the question, I've noticed that you are maybe more concerned about iterates of balls. 

For this, it depends on what you want. For example, if you want, for any $\varepsilon$ a ball of smaller radius that has iterates with volume close to the manifold, it is easy to get (say, for example a north-south map). I believe this should be not what you are looking for.

If you look for every ball at a time, this won't be true in general, and let me rephrase your question (or at least the one I am responding to): Is there a homeomorphism $f$ such that given a ball $B$ and $\varepsilon>0$ there exists an integer $n$ such that $f^n(B)$ has measure larger than $1-\varepsilon$.

If $f$ is not transitive (or at least, if it has at least two chain recurrence classes), then, there exists an open set such that $\overline{U} \subset f(U)$ so, if you consider a ball in $\overline{U}^c$ it will never reach measure bigger than the complement of $U$.

For $f$ transitive, there are also examples where this does not hold. For example, for conservative $f$, it trivially does not hold.

In other cases, I would presume that it is not true (at least for ``tipical'' diffeomorphisms), however, I don't know a proof nor an example.

Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure.

Sorry if this has nothing to do, but from how I understood the question, at least this should be useful.

Not sure if it is related, but in Lemma 1 of this paper (A.Avila, J. Bochi, A C1 generic map has no invariant absolutely continuous probability measure) it is proved that if a map has no invariant absolutely continuous probability measure, then, there exists a compact set $K$ of measure abitrarily close to $1$ which has an iterate with arbitrarily small measure.

Sorry if this has nothing to do, but from how I understood the question, at least this should be useful.

EDIT: Re-reading the question, I've noticed that you are maybe more concerned about balls. For this, it depends on what you want. For example, if you want, for any $\varepsilon$ a ball of smaller radius that has iterates with volume close to the manifold, it is easy to get (say, for example a north-south map). However, if you look for every ball at a time, this won't be true in general, for, example, if $f$ is not transitive, you get a reppeling neighborhood, so you won't aproach the total volume never. If $f$ is transitive, I believe you can manage to prove the same (that some balls won't increase their volume arbitrarily close to that of the manifold) but I have not an argument at the moment (it is clear if it is volume preserving, and otherwise, you will get some regions which are kind of ``repelling'').