Can anyone tell me what I can add to Buttercream frosting that doesn't include shortening and will still allow me to keep the cake at room temperature?
I hate the taste of shortening
There are several different types of icing that are referred to as buttercream, none of which require the use of shortening, including:
American Buttercream -- Butter, powdered sugar, perhaps some milk, and flavoring such as vanilla beaten together. While some recipes call for shortening, using actual butter gives a better flavor. See a sample recipe from Savory Sweet Life.
French Buttercream -- Egg yolks are beaten and cooked by adding hot sugar syrup (at the softball stage). The yolk mixture is then beaten until it is cool, and butter and then butter and flavoring is beaten in. See sample recipe from Chicago Tribune.
Italian Buttercream -- An Italian Meringue (egg whites beaten with hot sugar syrup) is prepared and then cooled, and butter and flavoring beaten in. See sample recipe from Martha Stewart.
Any of these are stable at room temperature for a day or two, but like most perishable foods, should not be held indefinitely.
In addition to buttercreams, you may wish to consider other frostings which don't contain shortening, including ganache (chocolate melted with hot cream and cooled), whipped ganache (ganache that has been beaten until it is foamy), and Seven Minute Frosting (essentially a meringue frosting).
Buttercream frosting is kept in the fridge because the butter softens too much at room temperature. Decorations lose their definition, and many people prefer the firmer texture for the taste.
There is no way to change the properties of butter, so you will have to work with another fat if you can't refrigerate. The problem is that all solid fats bring their own taste into the frosting. You can have frosting which tastes of butter and is safe to eat for 1-2 days outside of the fridge, but turns soft. (If that's what you want, SAJ's answer is great). You can have frosting which tastes like something else and keeps firm. But you can't have both at once.
If you want a firm frosting, shortening is usually the best choice, because it has the most neutral taste. But if you are for some reason sensitive to the taste, you can use other firm fats, which will change the taste. A good choice which will probably harmonize well with many cakes is coconut fat. It is very firm at room temperature (unless you live in the tropics), and cocos taste is frequently associated with sweet pastries. Palm fat could be another alternative, but I guess that more people will have an aversion to an unexpected palm fat taste in cake than to an unexpected cocos taste in cake.
A different solution might be cocoa butter softened with dairy butter. Cocoa butter on its own is too hard, but carefully mixing it should give you a more spreadable result slightly reminiscent of soft white chocolate. The problem (beside the expense) is that no standard recipe for buttercream will work with such a mixture. If you haven't done standard chocolate work before (truffles, guanduja etc. from scratch), I wouldn't recommend going that way, as it is really very finicky. But it might be worth a try if you have previous experience in conditoring.