  We study the following elliptic problem involving slightly subcritical
non-power nonlinearity $$\left\{\begin{array}{lll} -\Delta u
=\frac{|u|^{2^*-2}u}{[\ln(e+|u|)]^\epsilon}\ \ &{\rm in}\ \Omega, \\[2mm] u= 0
\ \ & {\rm on}\ \partial\Omega, \end{array} \right.$$ where $\Omega$ is a
bounded smooth domain in $\mathbb{R}^n$, $n\geq 3$, $2^*=\frac{2n}{n-2}$ is the
critical Sobolev exponent, $\epsilon>0$ is a small parameter. By the finite
dimensional Lyapunov-Schmidt reduction method, we construct a sign changing
bubble tower solution with the shape of a tower of bubbles as $\epsilon$ goes
to zero.
