I'm SURE there are other things I'd use it for, but the only thing I can think of right now would be if we were to suddenly get a bill from the IRS saying we owed back taxes. The IRS scares me.
If a big, unexpected expense were to come up, the first thing to go would be all of the frivolous budget lines - entertainment, blow money, etc. I could pare down the grocery budget some...consolidate my daytime car trips to save on gas...Basically, do what I do everyday, but MORE.
Next, the snowball would go on hold and that money would go toward the emergency.
Since we run our budget about a month ahead, in the event of a REAL emergency, I could probably actually scrounge up two months' worth of snowball money at once, right away, if necessary.
I can't think of many immediate emergencies that would need more cash than that right away. Of course, the major expensive emergencies - (Let's say a meteor crashed through our roof and our homeowners' policy didn't cover meteor damage...I don't know if it does or not!

) - Well, throwing our BEF at it wouldn't make much of a difference. So instead, we'd use the money we had on hand from taking the steps I just listed and figure out how to mitigate the loss and damage and then put the major expense on the budget (either to save for or find another way to get it taken care of).
Like I said, though...If the IRS came a'knockin', it's be "Bye-bye, BEF".