To get them to disappear, I think you have to read all the older notifications. If you read the last one first, it'll probably disappear, unless it's one of the five newest, perhaps. I think you're stuck with those until you get new ones.

I also like to delete notifications after I read them @joparrott

In the drop down box for notifications there used to be an option to click on "mark all read" or something like that.

Strangely last night that option was back again but today it is gone.

@joparrott Are you using Windows and a Windows browser? It seems like different people are experiencing different things. Using Windows I see this. Do you see the same thing? If I click see all it takes me to a page where I can see them all and also mark as read there. Once marked as read they disappear.