I just can’t work away from my place. I am wasting a lot of time at Starbucks. I’ll move to a new place in one month, and I hope to do better then.

Let me describe a sequence of fields isomorphic to R via iterations of exp. The multiplication of each of them is the sum of the next one. I could vary the base, make it dependent of the index of the consecutive iteration but first let me keep it simple.

Let:

• e_-2  :=  -∞
• e_-1  :=  0   (thus  e_-1 = e^e_-2)
• e_n  :=  e^e_n-1   for every  n=0 1 …

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so that  e_n = eⁿ  for every  n=0 1 …. We have the standard addition and multiplication operations in

R = (e_-2; ∞)

namely

• #₀  :=  + : (e_-2; ∞)² → (e_-2; ∞)
• #₁  :=  * : (e_-2; ∞)² → (e_-2; ∞)

followed by more advanced admun (add-multiplication) operations:

#_n : (e_n-3; ∞)² → (e_n-3; ∞)

defined by:

exp(x) #_n exp(y)  :=  exp(x #_n-1 y)

for every  n = 2 3 …

Let  R_n  :=  (e_n-2); ∞)  for each  n=0 1…  Then

(R_n  #_n  #_n+1  e_n-1  e_n)

is a field, where:

• R_n  is the set of elements of the field;
• #_n  is the field’s addition operation (it’s commutative and associative);
• #_n+1  is the field’s multiplication operation (it’s commutative and associative);
• e_n-1  is the field addition’s neutral element (“zero”, even if it looks like something else )
• e_n  is the field multiplication’s neutral element (“one”, even if it looks like something else)

The Starbucks is about to close, I need to go (while Starbucks at Arborland is open 24h–should I write there before I move?).

Yes, I went home then, but now I am editing my entry (cheating? :-)).