**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!**