Although it isn't on the Starbucks coffee menu, you can order a dirty chai at Starbucks by asking to have a shot of espresso added to your chai. You can also make it a "double dirty chai" by asking for two shots of espresso.
Dirty Chai is one of the many ways in which chai spices are used as flavors in other edibles, like chai frosting or chai smoothies.
Other examples of drinks that combine coffee and tea include Hong Kong's Yuanyang Coffee-Tea and About.com's award-winning Chai-Coffee Cocktail.
A similar concept to the idea of chai as a flavor is the 'mocha' flavor profile, which originated in mocha lattes and has since spread to various types of unexpected mocha recipes such as mocha cake.
It is arguable that the popularity of Dirty Chai is due in part to its provocative name and the general association of coffee and espresso with social deviance.
To make your own dirty chais, check out these Dirty Chai Latte Recipes.