In mathematics, Steinhaus–Moser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus's polygon notation.
(a number n in a triangle)
means nⁿ.

(a number n in a square)
is equivalent with "the number n inside n triangles, which are all nested."

(a number n in a pentagon)
is equivalent with "the number n inside n squares, which are all nested."

etc.: n written in an (m+1)-sided polygon
is equivalent with "the number n inside n m-sided polygons, which are all nested."

Steinhaus only defined the triangle, the square, and a circle , equivalent to the pentagon defined above.

Steinhaus defined:

"mega" is the number equivalent to 2 in a circle:

"megiston" is the number equivalent to 10 in a circle:

use the functions square(x) and triangle(x)

let M(n,m,p) be the number represented by the number n in m nested p-sided polygons; then the rules are:

$M(n,1,3)\; =\; n^n$

$M(n,1,p+1)\; =\; M(n,n,p)$

and

mega = $M(2,1,5)$

moser = 

Note that

is already a very large number, since
=
square(square(2)) = square(triangle(triangle(2))) =
square(triangle(2²)) =
square(triangle(4)) =
square(4⁴) =
square(256) =
triangle(triangle(triangle(...triangle(256)...))) 256 triangles =
triangle(triangle(triangle(...triangle(256²⁵⁶)...))) 255 triangles =
triangle(triangle(triangle(...triangle(3.2 × 10⁶¹⁶)...))) 254 triangles =
...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

With the function $f(x)=x^x$ we have mega = $f^(256)\; =\; f^(2)$ where the superscript denotes a functional power, not a numerical power.

We have (note the convention that powers are evaluated from right to left):

M(256,2,3) = $(256^)^=256^$

M(256,3,3) = $(256^)^=256^=256^$≈$256^$

M(256,4,3) ≈ $$

M(256,5,3) ≈ $$

mega = $M(256,256,3)approx(256uparrow)^257$, where $(256uparrow)^$ denotes a functional power of the function $f(n)=256^n$.

$M(256,1,3)approx\; 3.23\; imes\; 10^$

$M(256,2,3)approx10^$ ($log\; \_\; 616$ is added to the 616)

$M(256,3,3)approx10^$ ($619$ is added to the $1.99\; imes\; 10^$, which is negligible; therefore just a 10 is added at the bottom)

$M(256,4,3)approx10^$

mega = $M(256,256,3)approx(10uparrow)^1.99\; imes\; 10^$, where $(10uparrow)^$ denotes a functional power of the function $f(n)=10^n$. Hence