I'm looking for a frosting I can leave on the counter (under a cover of some kind) for a few days.
What kind of frosting (for a cake or cupcakes) would stand up to this?
Ideally, I would like this to be something that I could make from scratch.
Whipped vegetable shortening with powdered sugar. I can make it up a week in advance, probably even longer, and keep it in an airtight container in a cool place. And it has the advantage that it's pure white (if you use imitation vanilla extract, or lemon extract), so you can get vibrant colors on whatever you're decorating.
If you make too much, you can then make flowers and let them dry out, then store those for a year or more.
Here's the recipe I use, that I got from Jane who taught two of the cake decorating classes I took (and who decorated cakes professionally):
4 lb powdered sugar
1 TB cream of tartar
1 1/4 c. of white shortening (crisco ... the sticks are easy to measure)
a pinch of salt
1 TB extract of your choice
3/4 to 1c. of water (maybe more depending on humidity)
Put all ingredients except the water into a mixer. Add 3/4 c. water, and slowly add more to make the consistancy of your choice. Mix on low speed 'til mixed, then on #6 or 8 for 6 to 7 minutes. Consistency should be "creamy" -- like cream cheese when soft. Stiff icing should be creamy also.
I've never had a buttercream frosting consisting of nothing but butter, powdered sugar and vanilla (and often cocoa or coloring) go bad. Even when it's got a little milk in it. There are two reasons for this. One, it doesn't last long enough. Hey! Where'd that last piece of cake go? Two, though I'm no expert, I think it is the high amount of sugar as Michael alluded to. I've had the infrequent cake with a piece or two that have hung around for a week without any problems other than a little drying.
Homemade? A buttercream frosting made with water instead of milk will last for a couple of days.
I imagine the canned stuff you can buy in a grocery store will last longer due to preservatives.
As an alternative you could just make whatever frosting you felt like, and keep it refrigerated until you needed to use it.
If you have a buttercream recipe that calls for milk, substitute it with water, and if it calls for butter substitute it with shortening. Shortening is generally kept on the shelf, so it doesn't need to be refrigerated.
Basically whatever ingredients you add in, if they generally need to be refrigerated, then it can't be left out.