  There is a well developed intersection theory on smooth Artin stacks with
quasi-affine diagonal. However, for Artin stacks whose diagonal is not
quasi-finite the notion of the degree of a Chow cycle is not defined. In this
paper we propose a definition for the degree of a cycle on Artin toric stacks
whose underlying toric varieties are complete. As an application we define the
Euler characteristic of an Artin toric stack with complete good moduli space -
extending the definition of the orbifold Euler characteristic. An explicit
combinatorial formula is given for 3-dimensional Artin toric stacks.
