unbounded functions that are Lebesgue integrable?
Yes, for example $\displaystyle f(x) = 1/\sqrt x$ on (0,1).
[If you want a proof that f is Lebesgue integrable, let $\displaystyle f_n(x) = \begin{cases}n&(0<x\leqslant1/n^2),\\1/\sqrt x&(1/n^2<x<1).\end{cases}$ Note that each f_n is integrable (with integral 2 – (1/n)) and that (f_n) increases to f, and use the monotone convergence theorem.]