Find sets $E_1, E_2,\dots$ of finite outer measure s.t. $E_k \searrow E$ and $\lim |E_k|_e > |E|_e$

The unit interval can be partitioned into countably many sets $B_k$ of outer measure 1. (In fact, much more is true; it can be partitioned into continuum many Bernstein sets, i.e., sets such that every uncountable closed set meets them all.) Let $E_n=\bigcup_{k>n}B_k$. These sets $E_n$ all have outer measure 1 and they converge to the empty set.

The following paper shows that one can partition a closed set of positive measure in any number $\kappa$ of unmeasurable sets with full outer and inner measure, as long as $2\leq\kappa\leq\mathfrak{c}$.

Alexander Abian, Partition of nondenumerable closed sets of reals, Czechoslovak Mathematical Journal, Vol. 26 (1976), No. 2, 207--210.

For the rest, proceed as in the answer of Andreas Blass.