Recently, Pinelis (2022) gave careful consideration that large deviations of the sums of independent and identically distributed random variables with power-like tails of index $\alpha\in(0,2)$ are mainly due to just one of the summands. In this article, we show a corresponding result for $\alpha=0$, which is constructed by super-heavy tailed distributions. The proof uses a property of slow variation investigated by Bojanic and Seneta (1971). It is applied to not only the log-Pareto distribution but also the distribution of the super-Petersburg game.