I believe the set is just the set of all (non-integrable) measurable functions with $\sigma$-finite support.
In particular, if the space in question is $\sigma$-finite (like $\mathbf R$ with Lebesgue measure), then all measurable functions are pointwise limits of integrable functions.
Edit: cleaned up a bit.
In one direction:
- Clearly it is enough to show that for nonnegative functions;
- Let $f$ be a nonnegative measurable function, and $S_n$ an increasing sequence of sets of finite measure such that $\mathrm{supp}f\subseteq \bigcup_nS_n$.
- Let $A=\lbrace x\mid f(x)=\infty\rbrace$, $A_n=\lbrace x\mid f(x).
- Then for each $n$ put $f_n(x)=f(x)$ on $A_n\cap S_n$, $f_n(x)=n$ on $A\cap S_n$, $f_n(x)=0$ otherwise.
- Then $f_n$ are integrable and $f_n\to f$ pointwise.
In the other direction, we show that the support of a pointwise limit of a sequence of integrable function has $\sigma$-finite support:
- Take an arbitrary sequence of integrable functions $f_n$
- Any integrable function $f_n$ has a $\sigma$-finite support $B_n=\bigcup_m\lbrace x\mid \lvert f_n(x)\rvert>1/m\rbrace$.
- The support of the limit of $f_n$ is contained in $\bigcup_n B_n$ (because if at some point none of the functions is nonzero, neither is the limit), and hence $\sigma$-finite, so we're done.