An old chestnut goes like this:

A new restaurant opens near where a group of five friends live. These friends always eat together and they decide to periodically visit the new restaurant.

The restaurant serves some sort of foreign cuisine (I've seen the puzzle written with several different ones) with which none of the friends are familiar. On their first visit they find that they don't recognize anything on the menu so they order blindly, each of them ordering one of the dishes on the menu, not necessarily all distinct.

It is the custom in this culture for groups to share food, so the waiter simply puts the five dishes in the middle of the table.

After this first meal, the friends may be able to identify one or more of the dishes by its name (for instance, if two people ordered one meal and the others all ordered different meals, they will know the one they ordered two of, since they got two plates of it).

After three meals, the friends have tried all the items on the menu, and they can identify each of them by name. If there was one more item on the menu, they could not have done this, no matter how they ordered.

How many items were on the menu?

Note: There are variations of this problem for different numbers of friends and different numbers of visits, but this seems to be the original version.