The following question is exercise 134Xi in D. Fremlin's Measure Theory volume 1. It is a "Basic exercise". Usually, basic exercises are basic indeed and not too difficult. I stumble on this one, however.
Recall that a set $A \subset [0,1]$ has full outer measure if any (Lebesgue-)measurable $F \subset [0,1] \setminus A$ is negligible (i.e. the outer Lebesgue measure of $A$ is 1). Prove that there is a disjoint sequence $(A_n)$ of subsets of $[0,1]$, all of full outer measure.
Note that 134D (in the book quoted above) is the statement that there is some subset $C$ of the real line such that both $C$ and and its complement are of full outer measure. Of course we are trying to improve on this. I am not able to modify the proof of 134D in order to achive this, though.
[Also, I am aware of the fact that we can actually find a non-countable family of disjoint subsets of outer measure, but I would like to see an elementary (i.e. "basic") proof of the weaker result above.]