We consider the obstacle problem $$\mathrm{minimize}I(u)=\int_\Omega G(\nabla u)dx\mathrm{among functions}u:\Omega\rightarrow R\\ \mathrm{such that}u|_{\partial\Omega}= 0\mathrm{and}u\geq\Phi\mathrm{a.e.}$$ for a given function $\Phi\in C^2(\bar{\Omega}),\Phi|_{\partial\Omega} < 0$ and a bounded Lipschitz domain $\Omega$ in $\mathbf{R}^n$ . The growth properties of the convex integrand $G$ are described in terms of a $N$ -function $A:[0,\infty)\rightarrow[0,\infty)$ with $\overline{\lim_{t\rightarrow\infty}}A(t)t^{-2} < \infty$ . If $n\leq 3$ , we prove, under certain assumptions on $G,C^{1,\alpha}$ -partial regularity for the solution to the above obstacle problem. For the special case where $A(t)= t ln(1+ t)$ we obtain $C^{1,\alpha}$ -partial regularity when $n\leq 4$ . One of the main features of the paper is that we do not require any power growth of $G$ .