Rainfall events are very erratic at short and large temporal and spatial scales. The main objectives of the present study were (i) to describe different time series of daily precipitation records using both chaos theory and stable distribution methods and (ii) to search for potential relationships between chaotic and Lévy-stable parameters. We studied eight time series of daily rainfall from different latitudes around the world. Each rainfall signal spanned nine years (1997–2005). We used methods derived from chaos theory (embedding delays, spectrum of Lyapunov exponents, determinism tests and others) and parameters computed after fitting a stable distribution model to each differenced time series of rainfall data. All the rainfall signals showed chaotic structures with two positive Lyapunov exponents. The stability index was αb2 which accounts for the scale-free behavior of rainfall dynamics. There were found latent statistical relationships between chaotic and Lévy stable parameters. That represents a potential connection between chaotic behavior, sub-Gaussian statistics and rainfall dynamics. Future research should deal with the connection between chaotic invariants, stable parameters and rainfall phenomenology.