We prove a large deviation principle for the largest eigenvalue of Wigner matrices without Gaussian tails, namely such that the distribution tails $\mathbb{P}( |X_{1,1}|>t)$ and $\mathbb{P}(|X_{1,2}|>t)$ behave like $e^{-bt^α}$ and $e^{-at^α}$ respectively for some $a,b\in (0,+\infty)$ and $α\in (0,2)$. The large deviation principle is of speed $N^{α/2}$ and with a good rate function depending only on the tail distribution of the entries.