‎In this paper‎, ‎a new five-parameter so-called Beta-Gompertz Geometric (BGG) distribution is introduced that can have a decreasing‎, ‎increasing‎, ‎and bathtub-shaped failure rate function depending on its parameters‎. ‎Some mathematical properties of the this distribution‎, ‎such as the density and hazard rate functions‎, ‎moments‎, ‎moment generating function‎, ‎R and Shannon entropy‎, ‎Bonferroni and Lorenz curves and the mean deavations are provided‎. ‎We discuss maximum likelihood estimation of the BGG parameters from one observed sample‎. ‎At the end‎, ‎in order to show the BGG distribution flexibility‎, ‎an application using a real data set is presented‎.