In this paper, we study the following fourth order elliptic problem $(E_p)$:
\begin{eqnarray*}
(E_p) \left \{
\begin{array}{l}
\Delta^2 u = u^p \quad \mbox{in} \ \Omega, \\
u > 0 \quad \mbox{in} \ \Omega, \\
u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0
\end{array}
\right.
\end{eqnarray*}
where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and
$p >1$ is any positive number.
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We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$.
Combining the arguments of Ren-Wei [8] and Wei [10],
we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains,
they have one or two ``peaks'' away form the boundary.
If it happens that the only one peak point appears,
we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$
under the Navier boundary condition.