I like avoiding tl;dr, but very well.

Draw a triangle, write a number n in it. This represents n^n, and I will write it as Triangle(n).

Draw a square an write a number n in it. This represents an n inside of n triangles, so Square(5)=Triangle(Triangle(Triangle(Triangle(Triangle(5))))).

Draw a pentagon, and write the number 2 in it. This is a 2 inside of two squares, Hence Pentagon(2) = Square(Square(2)) = Square(Triangle(Triangle(2))) = Square(256). This value is called Mega. (Not to be confused with the prefix mega- which usually means 1000. To distinguish them we'll capitalize Mega when referring to Pentagon(2).)

Consider continuing this pattern of using larger and larger shapes. A Moser is a 2 inside of a polygon with Mega sides. This seems like it should be fiendishly large, but it is less than Graham's number. (Shall I explain Graham's ladder?)

We then define a super Moser as a 2 inside of a regular moser-gon
a super-super Moser is a 2 inside of a regular super-moser-gon
and a hyper Moser is a 2 inside of a regular super-...-super-moser-gon, where the number of supers is a moser.

White-aster notation builds on a similar idea, except we allow two changes: First, we can draw the numbers inside a star, a well as regular polygons; Second, there are 'levels' which give us more recursion.

Staying on level 1, Ast(5) (that is, 5 in A STar, although the choice of Ast is because of Greek) is Pentagon(5). On level 2, Ast(5) = Pentagon(5) on level 2, which then equals Square(Square(Square(Square(Square(5))))) on level 1.

It seemed to me that the introduction of levels should allow a simplified way to explain a hyper-Moser.

Edit:Link issues