We introduce the Gamma-Exponential Process (GEP), a prior over a large family of continuous time processes. A hierarchical version of this prior (HGEP; the Hierarchical GEP) yields a useful model for analyzing complex time series. Models based on HGEPs display many attractive properties: conjugacy, exchangeability and closed-form predictive distribution for the waiting times, and exact Gibbs updates for the time scale parameters. After establishing these properties, we show how posterior inference can be carried efficiently using Particle MCMC methods. This yields a MCMC algorithm that can resample entire sequences atomically while avoiding the complications of introducing slice and stick auxiliary variables. We applied our model to the problem of estimating the disease progression in Multiple Sclerosis, and to RNA evolutionary modeling. In both domains, we found that our model outperformed the standard rate matrix estimation approach.