No Shortening and No Refrigeration, Buttercream Frosting
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).