Combining extreme-value theory with Bayesian methods offers several advantages, such as the availability of posterior predictive inference or the ability to study irregular cases for frequentist statistics. When no prior information is available, objective Bayes aims at using an external rule to construct a prior distribution. In particular, we focus on the use of Jeffreys prior for the Poisson process characterization of extremes, a model which generalizes the two most frequent ones i.e. the extreme-value distribution (EVD) for block-maxima and the generalized Pareto distribution (GPD) for peaks-over-threshold. After showing posterior propriety results, we also compare different reparametrisations of the Poisson process to facilitate sampling by Markov chain Monte Carlo (MCMC). In particular, the influence of a hyperparameter of the model and the interest of parameters orthogonality for Bayesian inference are investigated on simulated data.