  We study the dual space of the variable Lebesgue space $\Lp$ with unbounded
exponent function $\pp$ and provide an answer to a question posed
in~[fiorenza-cruzuribe2013]. Our approach is to decompose the dual into a
topological direct sum of Banach spaces. The first component corresponds to the
dual in the bounded exponent case, and the second is, intuitively, the dual of
functions that live where the exponent is unbounded (in a heuristic sense). The
second space is extremely complicated, and we illustrate this with a series of
examples. In the special case of the variable sequence space $\ell^\pp$, we
show that this piece can be further decomposed into two spaces, one of which
can be characterized in terms of a generalization of finitely additive
measures. As part of our work, we also considered the question of dense subsets
in $\Lp$ for unbounded exponents. We constructed two examples, one for general
variable Lebesgue spaces $\Lp$ and one in the sequence space $\lp$. This gives
an answer to another question from~[fiorenza-cruzuribe2013].
