It is possible to accelerate growth of hierarchy if FGH will be nested in each step of this hierarchy. Let's define

,

,

iff is a limit ordinal.

Let that means is equal to for usual FGH with

(in other words , , iff is a limit ordinal).

In this case

where

and .

Example n=3: then ,

and .

For the calculation we must start new FGH (let it be to show difference) with

then

and

This way

and

After this we must overcome the limit ordinal (Indeed can be expressed as huge amount of iterations of some function )

and so on.

So if then (comparing with usual FGH with ) and that is why in this case . As I remember, it is named "catching ordinal" ( for this case)

To go further we can imagine for example

where for usual FGH with , where is Takeuti-Feferman-Buchholz ordinal - The supremum of the range of the Feferman theta function,

or even

where for usual FGH with , where is Rathjen's ordinal - The supremum of the range of the Rathjen's psi function (And, as I know, largest ordinal which is well-defined in professional math).

This way we can reach . Let is Roogol.

**Extensions**

Let's define

,

,

iff is a limit ordinal.

where is equal to for SFGH with

Let's name this hierachy as SFGH^1 and notate . After this same way we can define and and so on.

Let and Roogolplex is equal to