  Let $((0,1], T)$ be the doubling map in the unit interval and $\varphi$ be
the Saint-Petersburg potential, defined by $\varphi(x)=2^n$ if $x\in (2^{-n-1},
2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the
Birkhoff sum $S\_n(x)=\varphi(x)+\cdots+\varphi(T^{n-1}(x))$. With respect to
the Lebesgue measure, the Saint-Petersburg potential is not integrable and it
is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in
probability. We determine the Hausdorff dimension of the level set $\{x:
\lim\_{n\to\infty}S\_n(x)/n=\alpha\} \ (\alpha>0)$, as well as that of the set
$\{x: \lim\_{n\to\infty}S\_n(x)/\Psi(n)=\alpha\} \ (\alpha>0)$, when
$\Psi(n)=n\log n, n^a $ or $2^{n^\gamma}$ for $a>1$, $\gamma>0$. The fast
increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also
studied.
