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Here I will introduse you to a new function,that could grow pretty fast,but not extremely by googological standarts.I call it the iterating lambda function.It was more of a thought experiment,when I first thought of it.It's also very similar to the Ackerman function and the Sudan function.

Here are the basics to it:

\(\lambda_0(a,0)=a+1\)

\(\lambda_0(0,b)=\lambda_0(\lambda_0 \ldots (\lambda_0(\lambda_0(0,b-1),b-1),b-1),b-1)\ldots ),b-1)\) with \(b\) many \(b's\) from the center out.

\(\lambda_0(a,b)=\lambda_0(\lambda_0(a-1,b),b-1)\)


\(\lambda_0(a,b,c,\#,1)=\lambda_0(a,b,c,\#)\) (only if the array in the brackets is longer than 2)

\(\lambda_0(a,b,c)=\lambda_0(a,\lambda_0(a,b-1,c),c-1)\)

\(\lambda_0(a,1,c)=\lambda_0(a,\lambda_0(a,a,c-1),c-1)\)

\(\lambda_0(a,b,c,d)=\lambda_0(a,\lambda_0(a,b-1,c,d),c-1,d)\)

\(\lambda_0(a,b,1,d)=\lambda_0(a,b,\lambda_0(a,b-1,b,d),d-1)\)

\(\lambda_0(a,1,b,c)=\lambda_0(a,\lambda(a,a,b-1,c),b-1,c)\)

\(\lambda_0(a,b,c,d,e)=\lambda_0(a,\lambda_0(a,b-1,c,d,e),c-1,d,e)\)

\(\lambda_0(a,b,1,d,e)=\lambda_0(a,b,\lambda_0(a,b-1,b,d,e),d-1,e)\)

\(\lambda_0(a,b,c,1,e)=\lambda_0(a,b,c,\lambda_0(a,b,c-1,c,e),e-1)\)

And it keeps growing the same way!

\(\lambda_1(a)=\lambda_0\underbrace{(a,a,a,a,\ldots,a,a)}_a\)

\(\lambda_1(a,b)=\lambda_1(\lambda_1(a-1,b),b-1)\)

After that all rules applying to \(\lambda_0(a,b,c,\ldots)\) apply to \(\lambda_1(a,b,c,\ldots)\)

So do they for \(\lambda_n(a,b,c,\ldots)\)

And \(\lambda_n(a)=\lambda_{n-1}\underbrace{(a,a,a,a,\ldots,a,a,a)}_a\) 

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