Let $h$ be the indicator function of the rationals.  This is of Baire class 2, i.e. it is not the pointwise limit of a sequence of continuous functions, but it is the pointwise limit of a sequence of pointwise limits of continuous functions.
For example, let $r_1, r_2, \ldots$ be an enumeration of the rationals, 
$g_n(x)$ the indicator function of $S_n = \{r_1, r_2, \ldots, r_n\}$, and
$f_{ij}(x) = \exp(-i \;\text{dist}(x, S_j))$.