I can't think of an idiomatic way to express this exact request. If my friend is new to the office, I would probably need to explain that it takes two doses to fill the mug. Then on later occasions I could simply ask him to fill the mug or fill the mug up, or if he needs more explicit direction, to push the button twice or to put in two servings.
Since the coffee is being dispensed from a machine, you could ask for two shots (or a double shot) of coffee. This is not the usual use of shot in reference to beverages:
10a. a small amount of a drink, especially a strong alcoholic one
Shots is also used to refer to servings of espresso and wheatgrass and other strong non-alcoholic drinks. But since the serving size of those drinks is small (a shotglass holds about 1 fl. oz./ 30cc), asking for a shot of anything is an informal way to ask for a small portion of it, especially when it is dispensed from a pump. Someone might informally ask for a shot of whipped cream or a shot of mustard in this way.
You could also ask for a double. A double beverage contains twice the normal portion of its strongest component of the drink; a double latte is a café latte with two shots of espresso, a double martini has twice as much gin.
But again, until your friend becomes familiar with the machine and your mug, asking for a double or two shots will require some additional explanation, just the same as if you ask for two servings, or two measures, or two portions, or even two doses.