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Note: This belongs to Aarex Tiaokhiao.

Taken from https://sites.google.com/site/aarexnumbers/wiln

Up to ((N))

First, let define N = 1, NN = 2, and NNN...NNN with n N's = n

A line has 75 letters, so (N) = 75.

Then (N)N = 76, (N)NN = 77, (N)(N) = 75*2, etc.

Then (NN) = (N)(N)(N)...(N)(N)(N) with 75 (N)'s = 75^2

(NN)N = 75^2+1, (NN)(N) = 75*76, (NN)(NN) = 75*(75*2), etc.

We can have (NNN) = (NN)(NN)(NN)...(NN)(NN)(NN) with 75 (NN)'s, (NNNN) = (NNN)(NNN)(NNN)...(NNN)(NNN)(NNN) with 75 (NNN)'s, etc.

In that way, we reach 75^75, or 75^^2, or ((N)).

Up to [N] = (N)NN

We defined ((N)) = (NNN...NNN) with 75 N's.

Then ((N)N) = ((N))((N))((N))...((N))((N))((N)) with 75 ((N))'s.

Then ((N)(N)) = ((N)NNN...NNN) with 75 N's.

We continue: ((N)(N)(N)), ((NN)), ((NN)(NN)), ((NNN)), (((N))), etc.

That we reach 75^^75, or 75^^^2, or (((...(((N)))...))) with 75 levels, or [N], or (N)NN.

Up to (N)(N)

We define (N)NN, which is equal to (((...(((N)))...))) with 75 levels.

We define (NN)NN, which is equal to (N)NN(N)NN(N)NN ... (N)NN(N)NN(N)NN with 75 (N)NN's.

We continue with (NNN)NN, ((N))NN, ((N)N)NN, ((N)(N))NN, ((NN))NN, (((N)))NN, ((N)NN)NN, etc.

Then (N)NNN is equal to (((...(((N)NN)NN)NN...)NN)NN)NN with 75 levels.

Then (N)NNNN is equal to (((...(((N)NNN)NNN)NNN...)NNN)NNN)NNN with 75 levels.

And in general, (N)NNN...NNN is equal to (((...(((N)NNN...NN)NNN...NN)NNN...NN...)NNN...NN)NNN...NN)NNN...NN with 75 levels.

That we reach (N)(N).

Up to first fixed point of (N)

We have (N)_(N), it's equal to (N)_NNN...NNN with 75 N's.

Then (N)_(N)N = (((...(((N)(N))(N))(N)...)(N))(N))(N) with 75 levels.

Then we can have (N)_(N)NN, (N)_(N)(N), (N)_(NN), (N)_((N)), (N)_[N], (N)_(N)_(N), etc. Limit is (N)_(N)_(N)_..._(N)_(N)_(N).

BEYOND!

We define A = (N)_(N)_..._(N) with 75 levels

Then AA = A(N)_(N)_..._(N) with 75 levels.

Then (A) = AAA...AAA with 75 A's, etc.

Then B = (A)_(A)_..._(A) with 75 levels, C = (B)_(B)_..._(B) with 75 levels, etc.

But A = N_NN, B = N_NNN, C = N_NNNN, etc. And limit is N_N_N_..._N_N_NN.

THE END!

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