  Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on
$\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the
maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq
H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|. \] In this very general
context one of the most important results is the P\'olya-Vinogradov bound \[
M(t)\leq \left\|K\right\|_{\infty}\log 3p. \] where
$K:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of
$t$. In this paper we provide a lower bound for incomplete Kloosterman sum,
namely we prove that for any $\varepsilon >0$ there exists some
$a\in\mathbb{F}_{p}^{\times}$ such that \[ M(e(\tfrac{ax+\overline{x}}{p}))\geq
\Big(\frac{1-\varepsilon}{\sqrt{2}\pi}+o(1)\Big)\log\log p. \] Moreover we also
provide some result on the growth of the moments of
$\{M(e(\tfrac{ax+\overline{x}}{p}))\}_{a\in\mathbb{F}_{p}^{\times}}$.
